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        <div id="content-top1">Page 21 - 11-Math-3  Matrices and Determinants</div>
        <div id="content-top2">P. 21</div>
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    <pre style="white-space:pre-line"><code>31.. MQautaridcreastiacndEqDueatetiromninsants                                                           eeLLeeaarrnn..PPuunnjjaabb

  ï£®1Ã— 2 + (-1) Ã—1 + 0 Ã— 3                          1Ã— 0 + (-1) Ã— 4 + 0 Ã— 0     1Ã—1 + (-1) Ã— 2 + 0 Ã— 6 ï£¹
= ï£¯ï£¯2 Ã— 2 + 3Ã—1 + (-1) Ã— 3                         2 Ã— 0 + 3Ã— 4 + (-1) Ã— 0     2 Ã—1 + 3Ã— 2 + (-1) Ã— 6ï£ºï£º
                                                   1Ã— 0 + (-2) Ã— 4 + 3Ã— 0      1Ã—1 + (-2) Ã— 2 + 3Ã— 6ï£ºï£»
  ï£¯ï£°1Ã— 2 + (-2) Ã—1 + 3Ã— 3
                                                                                              (2)
ï£®1 -4 -1ï£¹
= ï£¯ï£¯4 12                  ï£º
                       2  ï£º

ï£°ï£¯9 -8 15ï£»ï£º

Thus from (1) and (2), AB â‰  BA

Note: Matrix multiplication is not commutative in general

                ï£® 2 -1 3 0 ï£¹
                ï£¯               0 4 -2ï£ºï£º ,then find AAt and (At).
Exam=ple 2: If  ï£¯      1

                ï£¯ï£°-3 5 2 -1ï£ºï£»

Solution : Taking transpose of A, we have

                          ï£® 2 1 -3ï£¹
                          ï£¯ï£¯-1                     ï£º
                          ï£¯3       0            5  ï£º
                At     =           4            2  ï£º  ,  so

                          ï£¯  0     -2 -1ï£ºï£»
                          ï£°

                          ï£®2       -1           3     0ï£¹     ï£®2    1   -3ï£¹
                                   0            4     -2ï£ºï£º   ï£¯ï£¯-1  0        ï£º
=AAt                      ï£¯  1     5            2     -1ï£»ï£º         4     5  ï£º
                          ï£¯                                        -2
                          ï£°ï£¯-3                               ï£¯3          2ï£º
                                                             ï£¯         -1ï£»ï£º
                                                             ï£°  0

                       ï£® 4+1+9+0                         2 + 0 +12 + 0      -6 - 5 + 6 + 0ï£¹
                       ï£¯                                 1 + 0 +16 + 4      -3 + 0 + 8 + 2ï£ºï£º
                    =  ï£¯  2  +  0  +  12        +  0     -3 + 0 + 8 + 2     9 + 25 + 4 +1ï£»ï£º

                       ï£¯ï£°-6 - 5 + 6 + 0

                       ï£®14 14 -5ï£¹
                    = ï£¯ï£¯14                         ï£º
                                   21           7  ï£º

                       ï£°ï£¯-5 7 39ï£ºï£»

                                                                                                         version: 1.1

                                                                       21</code></pre>
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