Suman Cole

2021-09-04

Solve each problem and simplify it with no variables
$A=\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$
what is $A+{B}^{2}$?
What is ${A}^{2}-B$?

Arnold Odonnell

Step 1
We have to find the matrix
$A=\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$
${A}^{2}=\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right]\left[\begin{array}{cc}a& -2\\ 1& 3\end{array}\right]$
$=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& -2+9\end{array}\right]=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& 7\end{array}\right]$
${A}^{2}-B=\left[\begin{array}{cc}{a}^{2}-2& -2a-6\\ a+3& 7\end{array}\right]-\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$
$=\left[\begin{array}{cc}{a}^{2}-6& -2a-1\\ a+3-b& 0\end{array}\right]$
${B}^{2}=\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]\left[\begin{array}{cc}4& -5\\ b& 7\end{array}\right]$
$=\left[\begin{array}{cc}16-5b& -20-35\\ 4b+7b& -5b+49\end{array}\right]$
$A+{B}^{2}=\left[\begin{array}{cc}a+16-5b& -57\\ 11b+1& -5b+52\end{array}\right]$
$A+{B}^{2}=\left[\begin{array}{cc}a+16-5b& -57\\ 11b+1& -5b+52\end{array}\right]$

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