Question

If $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix}$ and $\text{adj } A = \begin{bmatrix} 5 & x & -2 \\ 1 & 1 & 0 \\ -2 & -2 & y \end{bmatrix}$, then the value of $x + y$ is

• A. 6
• B. 3
• C. 4
• D. 5

Hint:

To avoid calculating the entire adjoint matrix, remember the index swap rule: The element at $(row, col)$ in the adjoint is always the cofactor of $(col, row)$ from the original matrix!

Answer 1

The Correct Option is D

Step 1: Recall how adjoint entries work.
The entry in row $i$, column $j$ of $\text{adj }A$ equals the cofactor of the element in row $j$, column $i$ of $A$. So we read positions in a crossed way.

Step 2: Find $x$.
Here $x$ sits at row 1, column 2 of $\text{adj }A$, so it is the cofactor of $a_{21}$. Delete row 2 and column 1 of $A$ and attach the sign $(-1)^{2+1}$: \[ x = -\begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix} = -(0-4) = 4. \]

Step 3: Find $y$.
Now $y$ is at row 3, column 3, so it is the cofactor of $a_{33}$, with sign $(-1)^{3+3}=+1$: \[ y = +\begin{vmatrix} 1 & 0 \\ -1 & 1 \end{vmatrix} = 1-0 = 1. \]

Step 4: Add them.
So $x+y = 4+1 = 5$. \[ \boxed{x+y = 5} \]