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:
We are given a $2 \times 2$ matrix $A$ containing unknown variables $a$ and $b$. We are also provided an algebraic matrix identity: $A \cdot \text{adj } A = A \cdot A^T$. We need to evaluate the scalar expression $5a + b$.

Step 2: Key Formula or Approach:
Recall the standard property of matrix systems involving the adjoint: $$A \cdot \text{adj } A = |A| \cdot I$$ where $|A|$ is the determinant of matrix $A$ and $I$ is the identity matrix. Thus, the given equation simplifies from $A \cdot \text{adj } A = A \cdot A^T$ to: $$|A| \cdot I = A \cdot A^T$$ Alternatively, since $A$ is invertible ($|A| \neq 0$), we can pre-multiply by $A^{-1}$ on both sides of $A \cdot \text{adj } A = A \cdot A^T$ to get $\text{adj } A = A^T$. Let's use this highly direct approach!

Step 3: Detailed Explanation:
Given: $$A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$$ The transpose of matrix $A$, denoted $A^T$, is formed by swapping rows and columns: $$A^T = \begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix}$$ For any $2 \times 2$ matrix $\begin{bmatrix} w & x \\ y & z \end{bmatrix}$, its adjoint is found by swapping the diagonal entries and changing the signs of the off-diagonal entries: $$\text{adj } A = \begin{bmatrix} 2 & b \\ -3 & 5a \end{bmatrix}$$ According to our simplified relation $\text{adj } A = A^T$: $$\begin{bmatrix} 2 & b \\ -3 & 5a \end{bmatrix} = \begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix}$$ By equating corresponding components of these two equal matrices, we generate a system of simple constraints: 1. From position $(1,1)$: $2 = 5a \implies 5a = 2$ 2. From position $(1,2)$: $b = 3$ 3. From position $(2,1)$: $-3 = -b \implies b = 3$ (consistent) 4. From position $(2,2)$: $5a = 2$ (consistent) We need to evaluate the value of the expression $5a + b$: $$5a + b = 2 + 3 = 5$$

Step 4: Final Answer:
The value of $5a + b$ is equal to 5, which matches option (D).