Question

$\mathrm{Let}A+2B=\left(\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]\mathrm{and}2A-B=\left(\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right],\mathrm{then}\mathrm{find}\mathrm{tr}\left(A\right)-\mathrm{tr}\left(B\right).$

• A. 3
• B. -1
• C. 2
• D. -3

Answer 1

The correct option is C 2

Here to find the value of tr(A) - tr(B), we need not to find the value of A and B.
We can find tr(A) - tr(B) using the properties of trace of matrix,i.e
tr(A) means$\to$sum of principal diagonal elements
$A+2B=\left(\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$

tr(A + 2B) = (1 - 3 + 1) = -1
or
tr(A) + 2tr(B) = -1 $\mapsto$(1)

2A - B =$\left(\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$

tr(2A - B) = (2 - 1 + 2) = 3 $\mapsto$2

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

tr(A) - tr(B) = 2

option (c) is correct