# Find matrices

Question:

Find matrices $A$ and $B$, if $A+B=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]$ and $A-B=\left[\begin{array}{ccc}-5 & -4 & 8 \\ 11 & 2 & 0 \\ -1 & 7 & 4\end{array}\right]$

Solution:

We get $(A+B)+(A-B)=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]+\left[\begin{array}{ccc}-5 & -4 & 8 \\ 11 & 2 & 0 \\ -1 & 7 & 4\end{array}\right]$

$2 \mathrm{~A}=\left[\begin{array}{ccc}-4 & -4 & 10 \\ 16 & 6 & -6 \\ 6 & 10 & 12\end{array}\right]$

$\mathrm{A}=\left[\begin{array}{ccc}-2 & -2 & 5 \\ 8 & 3 & -3 \\ 3 & 5 & 6\end{array}\right]$

Now Subtract (A-B) from (A+B)

$(\mathrm{A}+\mathrm{B})-(\mathrm{A}-\mathrm{B})=\left[\begin{array}{ccc}1 & 0 & 2 \\ 5 & 4 & -6 \\ 7 & 3 & 8\end{array}\right]-\left[\begin{array}{ccc}-5 & -4 & 8 \\ 11 & 2 & 0 \\ -1 & 7 & 4\end{array}\right]$

$(2 B)=\left[\begin{array}{ccc}6 & 4 & -6 \\ -6 & 2 & -6 \\ 8 & -4 & 4\end{array}\right]$

$B=\left[\begin{array}{ccc}3 & 2 & -3 \\ -3 & 1 & -3 \\ 4 & -2 & 2\end{array}\right]$

Conclusion: $\mathrm{A}=\left[\begin{array}{ccc}-2 & -2 & 5 \\ 8 & 3 & -3 \\ 3 & 5 & 6\end{array}\right], \mathrm{B}=\left[\begin{array}{ccc}3 & 2 & -3 \\ -3 & 1 & -3 \\ 4 & -2 & 2\end{array}\right]$