Let $A + 2B =\begin{bmatrix}1&2&0\\6&-3&3\\-5&3&1\end{bmatrix}$ and $2A-B=\begin{bmatrix}2&-1&5\\2&-1&6\\0&1&2\end{bmatrix}$, then find $tr(A) - tr (B)$.

2

Here to find the value of $tr (A)-tr(B)$, we need not to find the matrices A and B.

We can find $tr(4)-tr(B)$ using the properties of trace of matrix, i.e.,

$A + 2B =\begin{bmatrix}1&2&0\\6&-3&3\\-5&3&1\end{bmatrix}$

$⇒tr(A + 2B) = -1$

or $tr(A) + 2tr(B) = -1$    ....(1)

$2A-B=\begin{bmatrix}2&-1&5\\2&-1&6\\0&1&2\end{bmatrix}$

$⇒tr(2A-B) = 3$

or $2tr(A)-tr(B) = 3$   ...(2)

Solving (1) and (2), we get $tr(A) = 1$ and $tr(B) = - 1$

$⇒tr(A) - tr(B) = 2$