# Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA. A=begin{bmatrix}1 & -1 &15&0&-23&-2&2end{bmatrix} , B=begin{bmatrix}1 & 1 &01&-4&53&-1&2end{bmatrix}

Given matrix A and matrix B. Find (if possible) the matrices: (a) AB (b) BA.
$A=\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]$
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Step 1
(a) The given matrices are $A=\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]$
Step 2
Find AB
$AB=\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right]\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]$
$=\left[\begin{array}{ccc}1-1+3& 1+4-1& 0-5+2\\ 5+0-6& 5+0+2& 0+0-4\\ 3-2+6& 3+8-2& 0-10+4\end{array}\right]$
$=\left[\begin{array}{ccc}3& 4& -3\\ -1& 7& -4\\ 7& 9& -6\end{array}\right]$
Step 3
b) Find BA
$BA=\left[\begin{array}{ccc}1& 1& 0\\ 1& -4& 5\\ 3& -1& 2\end{array}\right]\left[\begin{array}{ccc}1& -1& 1\\ 5& 0& -2\\ 3& -2& 2\end{array}\right]$
$=\left[\begin{array}{ccc}1+5+0& -1+0+0& 1-2+0\\ 1-20+15& -1+0-10& 1+8+10\\ 3-5+6& -3+0-4& 3+2+4\end{array}\right]$
$=\left[\begin{array}{ccc}6& -1& -1\\ -4& -11& 19\\ 4& -7& 9\end{array}\right]$
Jeffrey Jordon