Question

The third element in the second row of adjoint of a matrix \(A=\left[a_{i j}\right]_{3 \times 3}\) (where \(a_{i j}=2 i+j\) ) is

• A. 2
• B. -2
• C. 4
• D. -4

Hint:

Build the matrix, then remember that the adjoint is the transpose of the cofactor matrix.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
\(\mathrm{adj}(A)_{23}=C_{32}\), the cofactor of the element in row 3, column 2 of \(A\).

Step 2: Build the Matrix
With \(a_{ij}=2i+j\): \(A=\begin{bmatrix}3&4&5\\5&6&7\\7&8&9\end{bmatrix}\).

Step 3: Cofactor
Delete row 3 and column 2: minor \(M_{32}=\begin{vmatrix}3&5\\5&7\end{vmatrix}=21-25=-4\).
\(C_{32}=(-1)^{3+2}(-4)=4\).

Step 4: Result
The required element is 4, option (C).

Final Answer:
The third element of the second row of the adjoint is 4. \[ \boxed{\text{(C) }4} \]