Mulailah

Tanpa permulaan, anda tidak akan sampai ke mana-mana.

Semangat

Semangat yang kuat mampu mengatasi apapun cobaan yang datang.

Konsisten

Lumbung emas dalam diri kamu adalah pikiran kamu. Kamu dapat menggalinya sedalam-dalamnya dan sepuas-puas yang kamu inginkan.

Pantang Menyerah

Gagal selepas usaha adalah hikmah, anda akan mendapat sesuatu yang lebih besar daripada apa yang anda sangkakan.

Be The One

Be the one is better than be the best.

Tuesday, January 24, 2012

Current State of Mobile Learning

JOHN TRAXLER ( UNIVERSITY OF WOLVERHAMPTON ; UNITED KINGDOM)

Since the start of the current millennium, experience and expertise in the development and delivery of mobile learning have blossomed and a community of practice has evolved that is distinct from the established communities of “tethered” e-learning. This community is currently visible mainly through dedicated international conference series, of which MLEARN is the most prestigious, rather than through any dedicated journals. So far, these forms of development and delivery have focused on short-term small-scale pilots and trials in the developed countries of Europe, North America, and the Pacifi c Rim, and there is a taxonomy emerging from these pilots and trials that suggests tacit and pragmatic conceptualisations of mobile learning. What has, however, developed less confi dently within this community is any theoretical conceptualisation of mobile learning and with it any evaluation methodologies specifically aligned to the unique attributes of mobile learning. Some advocates of mobile learning attempt to defi ne and conceptualize it in terms of devices and technologies; other advocates do so in terms of the mobility of learners and the mobility of learning, and in terms of the learners’ experience of learning with mobile devices.
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Monday, January 23, 2012

Inilah Planet-planet Layak Huni

TEMPO.CO, Jakarta - Lebih dari 600 planet ekstrasolar telah ditemukan. Beberapa di antaranya memiliki kondisi yang sangat cocok untuk tempat persemaian kehidupan. Berikut beberapa planet yang mungkin menopang makhluk hidup.

1. Gliese 581d
Waktu penemuan: 24 April 2007
Massa: 5,6 massa Bumi
Periode orbit: 67 hari
 
Gliese 581d

Dari sudut pandang kosmis, planet yang terletak 20 tahun cahaya dari Bumi ini amat dekat. GLiese 581d memiliki posisi istimewa karena terletak di tepian zona layak huni bintang induknya, sebuah bintang katai merah.

Planet ini cukup hangat dan basah, sehingga menjadi lahan yang tepat untuk bersemainya kehidupan. Atmosfer karbon tebal mengelilingi planet ini dan memberikan perlindungan bagi permukaannya.

2. Gliese 581g
Waktu penemuan: 29 September 2010
Massa: 3,1 massa Bumi
Periode orbit: 36 hari
Bumi dan Gliese 581g

 Saat ditemukan tahun lalu Gliese 581g tampak sebagai kandidat terbaik untuk planet mirip Bumi. Planet ini berada dalam sistem yang sama dengan Gliese 581d yang telah diketahui sebelumnya. Planet bermassa tiga kali Bumi ini berada di dalam zona layak huni dan mungkin memiliki atmosfer.

Penelitian lebih lanjut menunjukkan jejak yang terekam pada penelitian sebelumnya bukan berasal dari planet baru. Karena itu diperkirakan planet ini mungkin tidak pernah ada.

3. GJ 1214b
Waktu penemuan: 16 Desember 2009
Massa: 6,5 massa Bumi
Periode orbit: 1,5 hari
 
GJ 1214b

Planet ini disebut-sebut sebagai "dunia air", dan terletak 42 tahun cahaya dari bumi. Sebuah bintang katai merah bermassa seperlima massa matahari bercokol di jantung sistem, memberikan energi bagi planet ini.

Keunikan planet ini adalah seluruh permukaannya digenangi oleh air. Permukaan air dikelilingi oleh atmosfer beruap yang tebal dan awan hidrogen. Perhitungan terhadap unsur kimia menunjukkan planet ini mungkin ditinggali kehidupan. Massa planet sebesar 6,5 kali Bumi dengan diameter 2,7 kali lebih besar membuat planet ini disebut sebagai superbumi.

4. HD 209458b (Osiris)
Waktu penemuan: 5 November 1999
Massa: 218 massa Bumi
Periode orbit: 3,5 hari
 
HD 209458b

Planet ini terjarak 150 tahun cahaya dari Bumi. Atmosfer yang mengelilinginya diperkirakan mengandung uap air dan beberapa elemen organik yang juga dimiliki Bumi purba. Elemen ini menjadi adonan penting dalam pembentukan kehidupan.

Sayangnya, planet ini sangat dekat dengan bintang induknya, sehingga terlalu panas untuk ditinggali. Pun dengan kondisi planet yang berupa gas, sehingga tidak memiliki daratan.

5. Kepler-10b
Waktu penemuan: 10 Januari 2011
Massa: 4,5 massa Bumi
Periode orbit: 20 jam

Kepler-10b 
Andai terletak sedikit lebih jauh dari bintang induk, Kepler-10b bisa saja menyimpan kehidupan. Planet yang ditemukan pada 2011 ini pernah disebut sebagai bukti bahwa planet batuan seperti Bumi bisa tercipta di sekitar bintang lain. Karena itu Kepler-10b juga menjadi mata rantai penelitian ilmu keplanetan.

Elemen pendukung kehidupan seperti air dan atmosfer tak berlimpah di planet ini. Kondisi ini membuat planet ini sulit menjadi tempat munculnya kehidupan.

6. HD 85512b
Waktu penemuan: 17 Agustus 2011
Massa: 3,6 massa Bumi
Periode orbit: 54 hari

HD 85512b
 Planet yang baru ditemukan ini terletak 36 tahun cahaya dari bumi. Massanya yang cukup moderat, yaitu 3,6 kali massa Bumi, membuat planet batuan ini bisa menyimpan atmosfer. Penelitian lebih jauh mengungkap H2O, CO2, dan N2 bisa muncul di atmosfer planet.

Jarak planet ke bintang induk juga berada di tepian zona layak huni. Banyaknya kondisi menguntungkan ini membuat planet HD 85512b sebagai tempat paling mungkin menyimpan kehidupan.

6. Kepler-22b
Waktu penemuan: Desember 2011
Massa: 2,4 massa Bumi


Hampir semua karakteristik Bumi ada pada Kepler-22b. Tak mengherankan bila pakar astronomi menjuluki planet baru ini sebagai kembaran Bumi.

Planet ini, yang ditemukan observatorium antariksa NASA, Kepler, mengitari satu bintang, yang juga amat mirip dengan matahari kita. Bintang ini tergolong sebagai bintang kelas G yang sangat mirip dengan matahari. Begitu pula dengan jarak antara bintang dan planetnya
 DISCOVERY | ANTON WILLIAM
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ebook fisika

Fisika SMA kelas X
Fisika SMA kelas XI
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Sunday, January 22, 2012

Presidential Percentages



Each U.S. President’s name in the Choice Box has a number beside it. Your task is to calculate the answers and place the president’s name in the puzzle that corresponds with the problem number and the number beside the president’s name.


Across
1. 35% of 798
3. 14% of 654
6. 75% of 720
9. 25% of 125
10. 75% of 860
13. 80% of 204
15. 24% of 175
16. 12.5% of 96
17. 145% of 327



Down

1. 225% of 454
2. 87.5% of 900
4. 5% of 24
5. 10% of 720
7. 647% of 94
8. 60% of 81
11. 20% of 67
12. 84% of 84
14. 89% of 45
Read more »
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Saturday, January 21, 2012

Building Assembled in 6 Days

16-storey hotel build in six days

Watch as workers in China build The New Ark Hotel, breaking construction records by starting with an empty plot in Changsha and leaving the site with a 50m tall building in just 136 hours.
 
Developer Broad Sustainable Building, who worked until 10pm each night, say the five days and 16 hour construction time is thanks to the use of pre-fabricated parts.
The frame of the 50m building - which is built to withstand a magnitude nine earthquake - took just 46 hours to put together, with the building enclosure taking another 90.


Video




30-storey Chinese hotel built in 15 days

Chinese firm The Broad Group – the company that wowed the construction world last year when it build a 16-storey residential block in just five days – has done it again.
This time, the company has constructed a 30-storey, 17,000m2 hotel tower on the shores of the Dongting Lake in China’s Hunan Province in just 15 days.
The building has been designed to withstand magnitude 9 earthquakes and is claimed to be five times more energy efficient than standard towers, both in construction and materials used, and in the energy required to maintain and run the building.

Video


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Pasport - Jawapos 8 Agustus 2011


Setiap saat mulai perkuliahan, saya selalu bertanya kepada mahasiswa berapa orang yang sudah memiliki pasport. Tidak mengherankan, ternyata hanya sekitar 5% yang mengangkat tangan. Ketika ditanya berapa yang sudah pernah naik pesawat, jawabannya melonjak tajam. Hampir 90% mahasiswa saya sudah pernah melihat awan dari atas. Ini berarti mayoritas anak-anak kita hanyalah pelancong lokal.

Maka, berbeda dengan kebanyakan dosen yang memberi tugas kertas berupa PR dan paper, di kelas-kelas yang saya asuh saya memulainya dengan memberi tugas mengurus pasport. Setiap mahasiswa harus memiliki "surat ijin memasuki dunia global.". Tanpa pasport manusia akan kesepian, cupet, terkurung dalam kesempitan, menjadi pemimpin yang steril. Dua minggu kemudian, mahasiswa sudah bisa berbangga karena punya pasport.

Setelah itu mereka bertanya lagi, untuk apa pasport ini? Saya katakan, pergilah keluar negeri yang tak berbahasa Melayu. Tidak boleh ke Malaysia, Singapura, Timor Leste atau Brunei Darussalam. Pergilah sejauh yang mampu dan bisa dijangkau.
"Uang untuk beli tiketnya bagaimana, pak?"
Saya katakan saya tidak tahu. Dalam hidup ini, setahu saya hanya orang bodohlah yang selalu memulai pertanyaan hidup, apalagi memulai misi kehidupan dan tujuannya dari uang. Dan begitu seorang pemula bertanya uangnya dari mana, maka ia akan terbelenggu oleh constraint. Dan hampir pasti jawabannya hanyalah tidak ada uang, tidak bisa, dan tidak mungkin.

Pertanyaan seperti itu tak hanya ada di kepala mahasiswa, melainkan juga para dosen steril yang kurang jalan-jalan. Bagi mereka yang tak pernah melihat dunia, luar negeri terasa jauh, mahal, mewah, menembus batas kewajaran dan buang-buang uang. Maka tak heran banyak dosen yang takut sekolah ke luar negeri sehingga memilih kuliah di almamaternya sendiri. Padahal dunia yang terbuka bisa membukakan sejuta kesempatan untuk maju. Anda bisa mendapatkan sesuatu yang yang terbayangkan, pengetahuan, teknologi, kedewasaan, dan wisdom.

Namun beruntunglah, pertanyaan seperti itu tak pernah ada di kepala para pelancong, dan diantaranya adalah mahasiswa yang dikenal sebagai kelompok backpackers. Mereka adalah pemburu tiket dan penginapan super murah, menggendong ransel butut dan bersandal jepit, yang kalau kehabisan uang bekerja di warung sebagai pencuci piring. Perilaku melancong mereka sebenarnya tak ada bedanya dengan remaja-remaja Minang, Banjar, atau Bugis, yang merantau ke Pulau Jawa berbekal seadanya.Ini berarti tak banyak orang yang paham bahwa bepergian keluar negeri sudah tak semenyeramkan, sejauh, bahkan semewah di masa lalu.

Seorang mahasiswa asal daerah yang saya dorong pergi jauh, sekarang malah rajin bepergian. Ia bergabung ke dalam kelompok PKI (Pedagang Kaki Lima Internasional) yang tugasnya memetakan pameran-pameran besar yang dikoordinasi pemerintah. Disana mereka membuka lapak, mengambil resiko, menjajakan aneka barang kerajinan, dan pulangnya mereka jalan-jalan, ikut kursus, dan membawa dolar. Saat diwisuda, ia menghampiri saya dengan menunjukkan pasportnya yang tertera stempel imigrasi dari 35 negara. Selain kaya teori, matanya tajam mengendus peluang dan rasa percaya tinggi. Saat teman-temannya yang lulus cum-laude masih mencari kerja, ia sudah menjadi eksekutif di sebuah perusahaan besar di luar negeri.


The Next Convergence
Dalam bukunya yang berjudul The Next Convergence, penerima hadiah Nobel ekonomi Michael Spence mengatakan, dunia tengah memasuki Abad Ke tiga dari Revolusi Industri. dan sejak tahun 1950, rata-rata pendapatan penduduk dunia telah meningkat dua puluh kali lipat. Maka kendati penduduk miskin masih banyak, adalah hal yang biasa kalau kita menemukan perempuan miskin-lulusan SD dari sebuah dusun di Madura bolak-balik Surabaya-Hongkong.

Tetapi kita juga biasa menemukan mahasiswa yang hanya sibuk demo dan tak pernah keluar negeri sekalipun. Jangankan ke luar negeri, tahu harga tiket pesawat saja tidak, apalagi memiliki pasport.Maka bagi saya, penting bagi para pendidik untuk membawa anak-anak didiknya melihat dunia. Berbekal lima ratus ribu rupiah, anak-anak SD dari Pontianak dapat diajak menumpang bis melewati perbatasan Entekong memasuki Kuching. Dalam jarak tempuh sembilan jam mereka sudah mendapatkan pelajaran PPKN yang sangat penting, yaitu pupusnya kebangsaan karena kita kurang urus daerah perbatasan. Rumah-rumah kumuh, jalan berlubang, pedagang kecil yang tak diurus Pemda, dan infrastruktur yang buruk ada di bagian sini. Sedangkan hal sebaliknya ada di sisi seberang. Anak-anak yang melihat dunia akan terbuka matanya dan memakai nuraninya saat memimpin bangsa di masa depan. Di universitas Indonesia, setiap mahasiswa saya diwajibkan memiliki pasport dan melihat minimal satu negara.

Dulu saya sendiri yang menjadi gembala sekaligus guide nya. Kami menembus Chiangmay dan menyaksikan penduduk miskin di Thailand dan Vietnam bertarung melawan arus globalisasi. Namun belakangan saya berubah pikiran, kalau diantar oleh dosennya, kapan memiliki keberanian dan inisiatif? Maka perjalanan penuh pertanyaan pun mereka jalani. Saat anak-anak Indonesia ketakutan tak bisa berbahasa Inggris, anak-anak Korea dan Jepang yang huruf tulisannya jauh lebih rumit dan pronounciation-nya sulit dimengerti menjelajahi dunia tanpa rasa takut. Uniknya, anak-anak didik saya yang sudah punya pasport itu 99% akhirnya dapat pergi keluar negeri. Sekali lagi, jangan tanya darimana uangnya. Mereka memutar otak untuk mendapatkan tiket, menabung, mencari losmen-losmen murah, menghubungi sponsor dan mengedarkan kotak sumbangan. Tentu saja, kalau kurang sedikit ya ditomboki dosennya sendiri.

Namun harap dimaklumi, anak-anak didik saya yang wajahnya ndeso sekalipun kini dipasportnya tertera satu dua cap imigrasi luar negeri. Apakah mereka anak-anak orang kaya yang orangtuanya mampu membelikan mereka tiket? Tentu tidak. Di UI, sebagian mahasiswa kami adalah anak PNS, bahkan tidak jarang mereka anak petani dan nelayan. Tetapi mereka tak mau kalah dengan TKW yang meski tak sepandai mereka, kini sudah pandai berbahasa asing.
Anak-anak yang ditugaskan ke luar negeri secara mandiri ternyata memiliki daya inovasi dan inisiatif yang tumbuh. Rasa percaya diri mereka bangkit. Sekembalinya dari luar negeri mereka membawa segudang pengalaman, cerita, gambar dan foto yang ternyata sangat membentuk visi mereka.

Saya pikir ada baiknya para guru mulai membiasakan anak didiknya memiliki pasport. Pasport adalah tiket untuk melihat dunia, dan berawal dari pasport pulalah seorang santri dari Jawa Timur menjadi pengusaha di luar negeri. Di Italy saya bertemu Dewi Francesca, perempuan asal Bali yang memiliki kafe yang indah di Rocca di Papa. Dan karena pasport pulalah, Yohannes Surya mendapat bea siswa di Amerika Serikat. Ayo, jangan kalah dengan Gayus Tambunan atau Nazaruddin yang baru punya pasport dari uang negara.

Rhenald Kasali
Guru Besar Universitas Indonesia



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Wednesday, January 18, 2012

Worksheet

For IX Grade all about Algebra

Numbers


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Tuesday, January 17, 2012

Mobile and Personal Communication : Past, Present and Future


PAST
From the introduction of public mobile radio in the United States in 1946 until the first analog cellular system went into operation in Chicago in 1983,mobile radio systems were based on the trunking principle. In other words,the available frequency spectrum (in the 150 or 450 MHz band) was divided into a suitable number of frequency channels. A centralized, high power antenna was used to transmit signals to mobile receivers. Large mobile receivers were installed in automobiles (in the trunks), and the telephone sets also were rather large. A call originating from or terminating on a mobile terminal had to compete for one of the limited number of channels. The quality of service in terms of call blocking probabilities was very high—in the order of 20-25%. However, the users were willing to trade off the convenience of mobility against the poor quality of service in terms of call blocking and received signal quality. These systems were also severely limited in terms of capacity and coverage.

To alleviate the high blocking problem in the early systems, efforts were made to allow call originations from the mobile telephones to wait for a free channel. In the so-called automated mobile telephone system (AMTS), the mobile telephone user would key in the called number and press the send button. The receiver system would then start scanning for an idle channel by cycling through all the channels in the system. In some systems, the number of scan cycles was restricted, so that if an idle channel was not found within the allowed number of scans, the call would be blocked. However, incoming calls to mobile terminals (mostly originating from fixed terminals in public switched telephone networks) had no mechanism for awaiting a free channel and were blocked on all-channels-busy condition. Though the improvement in the quality of service in these systems was only marginal, they did provide some interesting performance modeling problems



PRESENT
Since the initial commercial introduction of advanced mobile phone system (AMPS) service in 1983, mobile communications has seen an explosive growth worldwide. Besides the frequency reuse capabilities provided by the cellular operation, advances in technologies for wireless access, digital signal processing, integrated circuits, and increased battery life have contributed to exponential growth in mobile and personal communication services. Systems are evolving to address a range of applications and markets, which include digital cellular, cordless telephony, satellite mobile, and paging and specialized mobile radio systems. Data capabilities of these systems are also coming into focus with the increasing user requirements for mobile data communications, driven by the need for e-mail and Internet access. Whereas the analog cellular mobile systems fall in the category of first-generation mobile systems, the digital cellular, low power wireless, and personal communication systems are now perceived as second-generation mobile/PCS systems.

The first digital cellular system specification was released in 1990 by the European Telecommunications (ETSI) for the global system for mobile communication (GSM) system. The GSM, DCS 1800 (1800 MHz version of GSM), and DECT (digital enhanced cordless telecommunications) systems developed by ETSI form the basis for mobile and personal communication services not only in Europe but in many other parts of the world including North America. The number of GSM subscribers worldwide exceeds 100 million and is growing rapidly.

FUTURE
To complement the cellular and personal communication networks, whose radio coverage will be confined to populated areas of the world (less than 15% of the earth's surface), a number of global mobile satellite systems are in advanced stages of planning and implementation. These systems are generally referred as global mobile personal communications by satellites (GMPCS).GMPCS systems like Iridium, Globalstar, and ICO use constellations of low earth orbit (LEO) or medium earth orbit (MEO) satellites and operate as overlay networks for existing cellular and PCS networks. Using dual-mode terminals, they will extend the coverage of cellular and PCS networks to any and all locations on the earth's surface. On the other hand, a LEO satellite system like Teledesic aims to provide high capacity satellite links to enable delivery of high bitrate and multimedia services to every location on the earth.
International Mobile Telecommunications -2000 (IMT-2000) is the standard being developed by the ITU to set the stage for the third generation of mobile communication systems. The IMT-2000 standard not only will consolidate under a single standard different wireless environments (cellular mobile, cordless telephony, satellite mobile services), but will also ensure global mobility in terms of global seamless roaming and delivery of services. ETSI is also developing a third-generation mobile communication system called Universal Mobile Telecommunication System (UMTS), which will belong to the family of IMT-2000 systems.
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Location: Dubai - United Arab Emirates

Monday, January 16, 2012

Matrix Operations



Before we give the formal definition of how to multiply two matrices, we will discuss an example from a real life situation. Consider a city with two kinds of population: the inner city population and the suburb population. We assume that every year 40% of the inner city population moves to the suburbs, while 30% of the suburb population moves to the inner part of the city. Let I (resp. S) be the initial population of the inner city (resp. the suburban area). So after one year, the population of the inner part is


0.6 I + 0.3 S


while the population of the suburbs is

0.4 I + 0.7 S


After two years, the population of the inner city is

0.6 (0.6 I + 0.3 S) + 0.3 (0.4 I + 0.7 S)


and the suburban population is given by

0.4 (0.6 I + 0.3 S) + 0.7(0.4 I + 0.7 S)

Blue Matrix 1 Myspace Layout 2.0

Is there a nice way of representing the two populations after a certain number of years? Let us show how matrices may be helpful to answer this question. Let us represent the two populations in one table (meaning a column object with two entries):

\begin{displaymath}\left(\begin{array}{c} I\\ S\\ \end{array}\right)\end{displaymath}


So after one year the table which gives the two populations is

\begin{displaymath}\left(\begin{array}{c} 0.6 I + 0.3 S\\ 0.4 I + 0.7 S\\ \end{array}\right)\end{displaymath}


If we consider the following rule (the product of two matrices)

\begin{displaymath}\left(\begin{array}{cc} a&b\\ c&d\\ \end{array}\right) \lef... ...left(\begin{array}{c} aI + bS\\ cI + dS\\ \end{array}\right),\end{displaymath}


then the populations after one year are given by the formula

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\right) \left(\begin{array}{c} I\\ S\\ \end{array}\right).\end{displaymath}


After two years the populations are

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\rig... ...ght) \left(\begin{array}{c} I\\ S\\ \end{array}\right)\Bigg).\end{displaymath}


Combining this formula with the above result, we get

\begin{displaymath}\left(\begin{array}{cc} 0.6&0.3\\ 0.4&0.7\\ \end{array}\rig... ...imes 0.4&0.4 \times 0.3 + 0.7 \times0.7\\ \end{array}\right). \end{displaymath}


In other words, we have

\begin{displaymath}\left(\begin{array}{cc} a&b\\ c&d\\ \end{array}\right) \lef... ...{array}{cc} ae+ bg&af+bh\\ ce + dg&cf+dh\\ \end{array}\right)\end{displaymath}

In fact, we do not need to have two matrices of the same size to multiply them. Above, we did multiply a (2x2) matrix with a (2x1) matrix (which gave a (2x1) matrix). In fact, the general rule says that in order to perform the multiplication AB, where A is a (mxn) matrix and B a (kxl) matrix, then we must have n=k. The result will be a (mxl) matrix. For example, we have

\begin{displaymath}\left(\begin{array}{ccc} a&b&c\\ d&e&f\\ \end{array}\right)... ...a +b\beta +c\nu\\ d\alpha +e\beta +f\nu\\ \end{array}\right).\end{displaymath}


Remember that though we were able to perform the above multiplication, it is not possible to perform the multiplication

\begin{displaymath}\left(\begin{array}{c} \alpha\\ \beta\\ \nu\\ \end{array}\... ...)\left(\begin{array}{ccc} a&b&c\\ d&e&f\\ \end{array}\right).\end{displaymath}


So we have to be very careful about multiplying matrices. Sentences like "multiply the two matrices A and B" do not make sense. You must know which of the two matrices will be to the right (of your multiplication) and which one will be to the left; in other words, we have to know whether we are asked to perform $A \times B$ or $B \times A$. Even if both multiplications do make sense (as in the case of square matrices with the same size), we still have to be very careful. Indeed, consider the two matrices

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\;\mb... ...nd}\; \left(\begin{array}{cc} 0&0\\ 1&0\\ \end{array}\right).\end{displaymath}


We have

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\left... ...ht) = \left(\begin{array}{cc} 1&0\\ 0&0\\ \end{array}\right) \end{displaymath}


and

\begin{displaymath}\left(\begin{array}{cc} 0&0\\ 1&0\\ \end{array}\right)\left... ...ht) = \left(\begin{array}{cc} 0&0\\ 0&1\\ \end{array}\right).\end{displaymath}


So what is the conclusion behind this example? The matrix multiplication is not commutative, the order in which matrices are multiplied is important. In fact, this little setback is a major problem in playing around with matrices. This is something that you must always be careful with. Let us show you another setback. We have

\begin{displaymath}\left(\begin{array}{cc} 0&1\\ 0&0\\ \end{array}\right)\left... ...egin{array}{cc} 0&0\\ 0&0\\ \end{array}\right);\;\mbox{i.e.},\end{displaymath}


the product of two non-zero matrices may be equal to the zero-matrix.




Properties involving Addition. Let A, B, and C be mxn matrices. We have
1.
A+B = B+A
2.
(A+B)+C = A + (B+C)
3.
$A + {\cal O} = A$
where $\cal O$ is the mxn zero-matrix (all its entries are equal to 0);
4.
$A+B = {\cal O}$ if and only if B = -A.

Properties involving Multiplication.

1.
Let A, B, and C be three matrices. If you can perform the products AB, (AB)C, BC, and A(BC), then we have

(AB)C = A (BC)


Note, for example, that if A is 2x3, B is 3x3, and C is 3x1, then the above products are possible (in this case, (AB)C is 2x1 matrix).
2.
If $\alpha$ and $\beta$ are numbers, and A is a matrix, then we have

\begin{displaymath}\alpha (\beta A) = (\alpha \beta) A\end{displaymath}


3.
If $\alpha$ is a number, and A and B are two matrices such that the product $A\cdot B$ is possible, then we have

\begin{displaymath}\alpha (AB) = (\alpha A)B = A (\alpha B)\end{displaymath}


4.
If A is an nxm matrix and $\cal O$ the mxk zero-matrix, then

\begin{displaymath}A {\cal O} = {\cal O}\end{displaymath}


Note that $A {\cal O}$ is the nxk zero-matrix. So if n is different from m, the two zero-matrices are different.

Properties involving Addition and Multiplication.
1.
Let A, B, and C be three matrices. If you can perform the appropriate products, then we have

(A+B)C = AC + BC


and

A(B+C) = AB + AC


2.
If $\alpha$ and $\beta$ are numbers, A and B are matrices, then we have

\begin{displaymath}\alpha (A+B) = \alpha A + \alpha B\end{displaymath}


and

\begin{displaymath}(\alpha +\beta)A = \alpha A + \beta B\end{displaymath}


Example. Consider the matrices



\begin{displaymath}A = \left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\right)... ...nd}\; C = \left(\begin{array}{ccc} 0&1&5\\ \end{array}\right).\end{displaymath}



Evaluate (AB)C and A(BC). Check that you get the same matrix.
Answer. We have



\begin{displaymath}AB = \left(\begin{array}{c} -1\\ -2\\ \end{array}\right)\end{displaymath}



so



\begin{displaymath}(AB)C = \left(\begin{array}{c} -1\\ -2\\ \end{array}\right)... ...t(\begin{array}{ccc} 0&-1&-5\\ 0&-2&-10\\ \end{array}\right).\end{displaymath}



On the other hand, we have



\begin{displaymath}BC = \left(\begin{array}{ccc} 0&2&10\\ 0&-1&-5\\ \end{array}\right)\end{displaymath}



so



\begin{displaymath}A(BC) = \left(\begin{array}{cc} 0&1\\ -1&0\\ \end{array}\ri... ...t(\begin{array}{ccc} 0&-1&-5\\ 0&-2&-10\\ \end{array}\right).\end{displaymath}




Example. Consider the matrices



\begin{displaymath}X = \left(\begin{array}{c} a\\ b\\ c\\ \end{array}\right)\... ...ray}{cccc} \alpha & \beta & \nu & \gamma\\ \end{array}\right).\end{displaymath}



It is easy to check that



\begin{displaymath}X = a \left(\begin{array}{c} 1\\ 0\\ 0\\ \end{array}\right... ...) + c \left(\begin{array}{c} 0\\ 0\\ 1\\ \end{array}\right) \end{displaymath}



and



\begin{displaymath}Y = \alpha \left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ \end{ar... ...ma \left(\begin{array}{cccc} 0 &0 & 0& 1\\ \end{array}\right).\end{displaymath}



These two formulas are called linear combinations. More on linear combinations will be discussed on a different page.
We have seen that matrix multiplication is different from normal multiplication (between numbers). Are there some similarities? For example, is there a matrix which plays a similar role as the number 1? The answer is yes. Indeed, consider the nxn matrix



\begin{displaymath}I_n = \left(\begin{array}{ccccc} 1&0&0&\cdots&0\\ 0&1&0&\cdo... ...cdot\\ \cdot&&&&\cdot\\ 0&0&0&\cdots&1\\ \end{array}\right).\end{displaymath}



In particular, we have



\begin{displaymath}I_2 = \left(\begin{array}{ccc} 1&0\\ 0&1\\ \end{array}\righ... ...egin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{array}\right).\end{displaymath}



The matrix In has similar behavior as the number 1. Indeed, for any nxn matrix A, we have



A In = In A = A



The matrix In is called the Identity Matrix of order n.
Example. Consider the matrices



\begin{displaymath}A = \left(\begin{array}{cc} 1&2\\ -1&-1\\ \end{array}\right... ...B = \left(\begin{array}{cc} -1&-2\\ 1&1\\ \end{array}\right).\end{displaymath}



Then it is easy to check that



\begin{displaymath}AB = I_2 \;\;\mbox{and}\;\; BA = I_2.\end{displaymath}




The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Indeed, for any nxm matrix A, we have



\begin{displaymath}I_n A = A\;\;\mbox{and}\;\; A I_m = A.\end{displaymath}



In particular, we have



\begin{displaymath}I_4 \left(\begin{array}{c} a\\ b\\ c\\ d\\ \end{array}\ri... ... \left(\begin{array}{c} a\\ b\\ c\\ d\\ \end{array}\right).\end{displaymath}
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