Question

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

• A. 7
• B. 9
• C. 13
• D. 5

Hint:

Always use the property $A(\text{adj } A) = |A|I$. Calculating the actual adjoint matrix and manually performing the multiplication is a massive and completely unnecessary waste of exam time.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a $2 \times 2$ matrix $A$ containing variables $a$ and $b$, and a matrix equation. We must solve for $a$ and $b$ using matrix multiplication properties, and then evaluate the expression $5a + b$.

Step 2: Key Formula or Approach:
1. A fundamental property of matrices is that $A \cdot (\text{adj } A) = |A| \cdot I$, where $|A|$ is the determinant and $I$ is the identity matrix.
2. Calculate the matrix multiplication $A \cdot A^T$ (where $A^T$ is the transpose).
3. Equate the two resulting matrices element-by-element to solve for the variables.

Step 3: Detailed Explanation:
First, find the determinant $|A|$:
$$|A| = (5a)(2) - (-b)(3) = 10a + 3b$$
So, the left hand side (LHS) is:
$$A \cdot \text{adj } A = \begin{bmatrix} 10a + 3b & 0
0 & 10a + 3b \end{bmatrix}$$
Next, define the transpose $A^T$:
$$A^T = \begin{bmatrix} 5a & 3
-b & 2 \end{bmatrix}$$
Calculate the right hand side (RHS), $A \cdot A^T$:
$$A A^T = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix} \begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix}$$
Perform the multiplication:
Row 1 $\times$ Col 1: $(5a)(5a) + (-b)(-b) = 25a^2 + b^2$
Row 1 $\times$ Col 2: $(5a)(3) + (-b)(2) = 15a - 2b$
Row 2 $\times$ Col 1: $(3)(5a) + (2)(-b) = 15a - 2b$
Row 2 $\times$ Col 2: $(3)(3) + (2)(2) = 9 + 4 = 13$
Equate the elements of the LHS and RHS matrices:
$$\begin{bmatrix} 10a + 3b & 0 \\ 0 & 10a + 3b \end{bmatrix} = \begin{bmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{bmatrix}$$
This gives us two simple linear equations to solve:
From the off-diagonal elements: $15a - 2b = 0 \implies 2b = 15a \implies b = \frac{15a}{2}$
From the bottom-right elements: $10a + 3b = 13$
Substitute $b$ into the second equation:
$$10a + 3\left(\frac{15a}{2}\right) = 13$$
$$10a + \frac{45a}{2} = 13 \implies \frac{20a + 45a}{2} = 13 \implies 65a = 26 \implies a = \frac{26}{65} = \frac{2}{5}$$
Now find $b$:
$$b = \frac{15(2/5)}{2} = \frac{6}{2} = 3$$
Evaluate the requested expression $5a + b$:
$$5\left(\frac{2}{5}\right) + 3 = 2 + 3 = 5$$

Step 4: Final Answer:
The value is 5, matching option (d).