# find the product of AB? A=begin{bmatrix}3 & 1 6 & 0 5&0 end{bmatrix} B=begin{bmatrix}1 & 5 & 3 -1 & 2 & -3 end{bmatrix} a) begin{bmatrix}2 & 17& 6 6 &

Chesley 2020-12-02 Answered
find the product of AB?
$A=\left[\begin{array}{cc}3& 1\\ 6& 0\\ 5& 0\end{array}\right]$
$B=\left[\begin{array}{ccc}1& 5& 3\\ -1& 2& -3\end{array}\right]$
a) $\left[\begin{array}{ccc}2& 17& 6\\ 6& 30& 18\\ 5& 25& 15\end{array}\right]$
b) $\left[\begin{array}{ccc}16& 6& 18\\ -2& 10& -30\\ 25& 17& 6\end{array}\right]$
c) $\left[\begin{array}{ccc}21& 6& -23\\ 2& 30& 5\\ -25& 12& 17\end{array}\right]$
d) The product is not defined
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## Expert Answer

Macsen Nixon
Answered 2020-12-03 Author has 117 answers
Step 1
Given matrices are:
$A=\left[\begin{array}{cc}3& 1\\ 6& 0\\ 5& 0\end{array}\right]$
and
$B=\left[\begin{array}{ccc}1& 5& 3\\ -1& 2& -3\end{array}\right]$
We have to find the value of product of A and B or AB.
In general if we have two matrices
$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$
$B=\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]$
We can multiply two matrices only when number of columns of first matrix is equal to number of rows of second matrix.
If the order of first matrix is $m×n$ then order of second matrix should be $n×p$.
Here order of first matrix is $2×2$ and order of second matrix is $2×2$
Therefore,
$AB=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]$
$=\left[\begin{array}{cc}ax+bz& ay+bw\\ cx+dz& cy+dw\end{array}\right]$ We do multiply row by column or column by row.
Step 2
Applying above rule in the given matrices, we get
Here, order of first matrix is $3×2$ and order of second matrix is $2×3$
So, matrices multiplication in this case is possible.
$AB=\left[\begin{array}{cc}3& 1\\ 6& 0\\ 5& 0\end{array}\right]\left[\begin{array}{ccc}1& 5& 3\\ -1& 2& -3\end{array}\right]$
$=\left[\begin{array}{ccc}3-1& 15+2& 9-3\\ 6-0& 30+0& 18-0\\ 5-0& 25+0& 15-0\end{array}\right]$
$=\left[\begin{array}{ccc}2& 17& 6\\ 6& 30& 18\\ 5& 25& 15\end{array}\right]$
Hence, first option is correct.
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Jeffrey Jordon
Answered 2022-01-22 Author has 2064 answers

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