Question

Let the matrix $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$ 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)

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