# A=begin{bmatrix}4 & 0 0 & -6 end{bmatrix} , B=begin{bmatrix}0 & -3 3 & -3 end{bmatrix} Find a matrix C which satisfies 5A+C=B Find a matrix D which satisfies A+B+D=I

$A=\left[\begin{array}{cc}4& 0\\ 0& -6\end{array}\right],B=\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]$
Find a matrix C which satisfies 5A+C=B
Find a matrix D which satisfies A+B+D=I
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Step 1
Given matrices are,
$A=\left[\begin{array}{cc}4& 0\\ 0& -6\end{array}\right],B=\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]$
Step 2
(1) Find a matrix C which satisfies
5A+C=B
C=B-5A
$C=\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]-5\left[\begin{array}{cc}4& 0\\ 0& -6\end{array}\right]$
Step 3
After solving
$C=\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]-\left[\begin{array}{cc}20& 0\\ 0& -30\end{array}\right]$
$C=\left[\begin{array}{cc}-20& -3\\ 3& 27\end{array}\right]$
Step 4
(2) Find a matrix D which satisfies A+B+D=I
Let $D=\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]\phantom{\rule{0ex}{0ex}}=\left[\begin{array}{cc}4& 0\\ 0& -6\end{array}\right]+\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]+\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}4& 0\\ 0& -6\end{array}\right]-\left[\begin{array}{cc}0& -3\\ 3& -3\end{array}\right]$
$\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]=\left[\begin{array}{cc}-3& 3\\ -3& 10\end{array}\right]$
Step 5
Hence the D will be
$D=\left[\begin{array}{cc}-3& 3\\ -3& 10\end{array}\right]$
Jeffrey Jordon