# Find a matrix A such that 2A – 3B + 5C = O, where $B = \begin{bmatrix} -2 & 2 & 0\\ 3 &1 &4 \end{bmatrix}$ and $C = \begin{bmatrix} 2 & 0 & 2\\ 7 &1 &6 \end{bmatrix}$

2A – 3B + 5C = O

Given:    $B = \begin{bmatrix} -2 & 2 & 0\\ 3 &1 &4 \end{bmatrix}$     and $C = \begin{bmatrix} 2 & 0 & -2\\ 7 &1 &6 \end{bmatrix}$

and 2A – 3B + 5C = O

2A $\Rightarrow$ -5C + 3B

$A = \frac{1}{2}[3B -5C]$

$= \frac{1}{2}\left[ 3\begin{bmatrix} -2 & 2 & 0\\ 3 &1 &4 \end{bmatrix} -5 \begin{bmatrix} 2 & 0 & -2\\ 7 &1 &6 \end{bmatrix}\right]$

$= \frac{1}{2}\left[ \begin{bmatrix} -6 & 6 & 0\\ 9 &3 &12 \end{bmatrix} - \begin{bmatrix} 10 & 0 & -10\\ 35 &5 &30 \end{bmatrix}\right]$

$= \frac{1}{2}\begin{bmatrix} -16 & 6 & 10\\ -26 &-2 &-18\end{bmatrix}$

$= \begin{bmatrix} -8 & 3 & 5\\ -13 &-1 &-9\end{bmatrix}$

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