Question

If \(A = [\begin{array}{cc}5a & -b \\ 3 & 2\end{array}]\) and \(A\,(\text{adj }A) = AA^T\), Then \(5a+b =\)

• A. \(2\)
• B. \(3\)
• C. \(5\)
• D. \(\frac{15}{2}\)

Hint:

A adj A equals the determinant times identity; compare entries with A A^T.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
\(A\,(\operatorname{adj}A) = |A|\,I\). Here \(|A| = 10a + 3b\).

Step 2: Compute A A^T:
\[ AA^T = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}\begin{bmatrix} 5a & 3 \\ -b & 2 \end{bmatrix} = \begin{bmatrix} 25a^2 + b^2 & 15a - 2b \\ 15a - 2b & 13 \end{bmatrix} \]

Step 3: Compare with (10a + 3b) I:
Off-diagonal: \(15a - 2b = 0\), so \(b = \tfrac{15a}{2}\).
Entry (2,2): \(13 = 10a + 3b\).
Substitute: \(10a + \tfrac{45a}{2} = \tfrac{65a}{2} = 13 \Rightarrow a = \tfrac25\), so \(b = 3\).
Check entry (1,1): \(25\cdot\tfrac{4}{25} + 9 = 13\). Correct.

Step 4: Compute:
\(5a + b = 2 + 3 = 5\). Options (A), (B) and (D) do not match.

Final Answer:
a = 2/5 and b = 3, so 5a + b = 5. \[ \boxed{\text{(C) }5} \]