Question

If $A=\left[\begin{array}{ccc}6& 10& 100\\ 7& 1& 0\\ 0& 9& 10\end{array}\right]$, then $Tr\left(A^T\right)=$
($Tr$ denotes trace of a matrix)

• A. $-17$
• B. $17$
• C. $-\frac{1}{17}$
• D. $\frac{1}{17}$

Answer 1

The correct option is B $17$

We know that the matrices $A$ and $A^T$ have same diagonal elements;
For example: $\left[\begin{array}{cc}5& 6\\ 2& 3\end{array}\right]$ has transpose $\left[\begin{array}{cc}5& 2\\ 6& 3\end{array}\right]$ both have same diagonal elements.
Hence the result holds:
$TrA=Tr{A}^T$ ( $\because$ diagonal elements remain the same for a matrix and its transpose)
$\therefore Tr{A}^T=\sum _{i=1}^n{a}_{ii}$
$\therefore Tr{A}^T=6+1+10$
$=17$.