Question

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \cdot \text{adj } A = A \cdot A^T$, then $5a + b =$

• A. 13
• B. 4
• C. $-1$
• D. 5

Hint:

For any matrix $A$, the equation $A \cdot \text{adj } A = A \cdot A^T$ simplifies instantly to $\text{adj } A = A^T$. For a $2 \times 2$ matrix, this means the off-diagonal elements swap signs and locations simultaneously, which lets you read off $b = 3$ and $5a = 2$ without writing down a single product!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Given matrix A and the identity A · adj A = A · Aᵀ, we must evaluate the scalar expression 5a + b.

Step 2: Key Formula or Approach:
For an invertible matrix, the identity A · adj A = A · Aᵀ simplifies to adj A = Aᵀ by pre-multiplying both sides with A⁻¹.

Step 3: Detailed Explanation:
For A = [[5a, –b], [3, 2]], Aᵀ = [[5a, 3], [–b, 2]] and adj A = [[2, b], [–3, 5a]]. Equating entries: 2 = 5a and b = 3, so 5a + b = 2 + 3 = 5.

Step 4: Final Answer:
The value of 5a + b is 5, matching option (D).