Question

Let $A+2B=\begin{bmatrix} 2 & 4 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 5 \end{bmatrix}$ and $2A-B=\begin{bmatrix} 6 & -2 & 4 \\ 6 & 1 & 5 \\ 6 & 3 & 4 \end{bmatrix}$ , then $tr\left(A\right)-tr\left(B\right)$ is equal to

(where, $tr\left(A\right)=$ trace of matrix $A$ i.e. sum of the principal diagonal elements of matrix $A$ )

Answer 1

Let, $x=\begin{bmatrix} 2 & 4 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 5 \end{bmatrix}$
$y=\begin{bmatrix} 6 & -2 & 4 \\ 6 & 1 & 5 \\ 2 & 3 & 4 \end{bmatrix}$
$A+2B=x$
$2A-B=y$
$A=\frac{1}{5}\left(x + 2 y\right),B=\frac{1}{5}\left(2 x - y\right)$
$A-B=\frac{1}{5}\left(- x + 3 y\right)=\frac{1}{5}\begin{bmatrix} 16 & -10 & 12 \\ 12 & 6 & 12 \\ 1 & 6 & 7 \end{bmatrix}$
$tr\left(A\right)-tr\left(B\right)=tr\left(A - B\right)$
$=\frac{29}{5}$