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: Understanding the Question:
We are given a $3 \times 3$ matrix $A$ and its adjoint matrix $\text{adj } A$ with two unknown elements, $x$ and $y$. We need to find the specific values of $x$ and $y$ to compute their sum.

Step 2: Key Formula or Approach:
The adjoint of a matrix $A$ is the transpose of its cofactor matrix.
This means the element at position $(i, j)$ in $\text{adj } A$ is exactly equal to the cofactor of the element at position $(j, i)$ in the original matrix $A$.
The cofactor $C_{ji}$ is calculated as $(-1)^{j+i} \times M_{ji}$, where $M_{ji}$ is the minor of the element $a_{ji}$.

Step 3: Detailed Explanation:
Find $x$:
$x$ is located at row 1, column 2 of $\text{adj } A$.
Therefore, $x$ is the cofactor of the element at row 2, column 1 of matrix $A$ ($a_{21}$).
$$x = C_{21} = (-1)^{2+1} \begin{vmatrix} 0 & 2 \\ 2 & 1 \end{vmatrix}$$ $$x = -(0 \times 1 - 2 \times 2) = -(0 - 4) = 4$$ Find $y$:
$y$ is located at row 3, column 3 of $\text{adj } A$.
Therefore, $y$ is the cofactor of the element at row 3, column 3 of matrix $A$ ($a_{33}$).
$$y = C_{33} = (-1)^{3+3} \begin{vmatrix} 1 & 0 \\ -1 & 1 \end{vmatrix}$$ $$y = +(1 \times 1 - 0 \times (-1)) = 1$$ Calculate the sum:
$$x + y = 4 + 1 = 5$$

Step 4: Final Answer:
The value of $x + y$ is 5, which matches option (D).