1

$A=\left[\begin{array}{ccc} 2&1&0\\0&3&-1 \end{array}\right]$

$B=\left[\begin{array}{cc}a&1\\1&b\\b&a\end{array}\right]$

Matrices

Find $AB$

peterwhy
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tess
  • 21
  • Please, use latex to write math. I don't have a clue of what are the matrices and what do you want, you just need the product AB?, is this homework?. – Cure Sep 24 '13 at 01:21
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    Welcome to math.SE! Please consider taking the time to read the [faq] to familiarise yourself with some of our common practices. In addition, this page should give you a start at learning how to typeset mathematics here so that your posts say what you want them to, and also look good. As this question appears to be homework, please consider reading this page for information about asking effective homework-related questions. Cheers! – ronno Sep 24 '13 at 01:33

2 Answers2

1

It seems that the matrix $A$ is $2\times 3$ and $B_{3\times 2}$ so we have $$AB=C_{2\times 2}$$ Now to do the latter matrix, you need to multiply the only row of $A$ by all columns of $B$ as follows:

 C[11]=[2 1 0][a  = 2*a+1*1+0*b=1+2a
               1
               b]

 C[12]=[2 1 0][1  = 2*1+1*b+0*a=2+b
               b
               a]

 C[21]=[0 3 -1][a  = 0*a+1*3+(-1)*b=3-b
                1
                b]

 C[22]=[0 3 -1][1  = 0*1+3*b+(-1)*a=3b-a
                b
                a]
Mikasa
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0

$$AB=\left[\begin{array}{ccc} 2&1&0\\0&3&-1 \end{array}\right]\left[\begin{array}{cc}a&1\\1&b\\b&a\end{array}\right]$$

If you know basic matrix multiplication, you should know

$$\begin{align} AB =& \left[\begin{array}{cc} A\left[\begin{array}{c}a\\1\\b\end{array}\right]& A\left[\begin{array}{c}1\\b\\a\end{array}\right] \end{array}\right]\\=& \left[\begin{array}{cc} \left[\begin{array}{ccc}2&1&0\\0&3&-1\end{array}\right]\left[\begin{array}{c}a\\1\\b\end{array}\right]& \left[\begin{array}{ccc}2&1&0\\0&3&-1\end{array}\right]\left[\begin{array}{c}1\\b\\a\end{array}\right] \end{array}\right]\\ =&\left[\begin{array}{cc} \left[\begin{array}{ccc}2&1&0\end{array}\right]\left[\begin{array}{c}a\\1\\b\end{array}\right]& \left[\begin{array}{ccc}2&1&0\end{array}\right]\left[\begin{array}{c}1\\b\\a\end{array}\right]\\ \left[\begin{array}{ccc}0&3&-1\end{array}\right]\left[\begin{array}{c}a\\1\\b\end{array}\right]& \left[\begin{array}{ccc}0&3&-1\end{array}\right]\left[\begin{array}{c}1\\b\\a\end{array}\right] \end{array}\right]\\ =&\left[\begin{array}{cc} 2a+1& \left[\begin{array}{ccc}2&1&0\end{array}\right]\left[\begin{array}{c}1\\b\\a\end{array}\right]\\ \left[\begin{array}{ccc}0&3&-1\end{array}\right]\left[\begin{array}{c}a\\1\\b\end{array}\right]& \left[\begin{array}{ccc}0&3&-1\end{array}\right]\left[\begin{array}{c}1\\b\\a\end{array}\right] \end{array}\right] \end{align}$$

For each value in the resultant array, the row vector and column vector multiplication is simply the sum of product of corresponding values. For example

$$\begin{align} \left[\begin{array}{ccc}2&1&0\end{array}\right]\left[\begin{array}{c}a\\1\\b\end{array}\right] =& 2\times a + 1\times 1 + 0\times b\\ =& 2a +1 \end{align}$$

You should be able to find all values in terms of $a$ and $b$ now.

peterwhy
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