Question

Let the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$ and $BA=A$ where $B$ represent $3\times 3$ order matrix. If the total number of $1$ in matrix $A^{-1}$ and matrix $B$ are $p$ and $q$ respectively, then the value of $p+q$ is equal to

• A. $3$
• B. $4$
• C. $5$
• D. $7$

Answer 1

Answer: (D)

$A^{-1}=\frac{1}{|A|}adj\left(A\right)\Rightarrow A^{-1}=\left[\begin{array}{ccc}1& -2& 1\\ 0& 1& -2\\ 0& 0& 1\end{array}\right]$
$\left|A\right|=1$ and $BA=A$
$BA{A}^{-1}=A{A}^{-1}\Rightarrow B=I$
Total number of $1$ in matrix $B=p=3$ and matrix $A^{-1}=q=4$
Hence, $p+q=7$