MHT CET · Maths · Matrices
If \(\mathrm{A}=\left[\begin{array}{cc}5 a & -\mathrm{b} \\ 3 & 2\end{array}\right]\) and A.adj \(\mathrm{A}=\mathrm{AA}^{\mathrm{T}}\), then \(5 a+\mathrm{b}=\)
- A 7
- B 9
- C 13
- D 5
Answer & Solution
Correct Answer
(D) 5
Step-by-step Solution
Detailed explanation
\( \mathrm{A} \cdot \operatorname{adj} \mathrm{A} = \det(\mathrm{A}) \mathrm{I} = ((5a)(2) - (-b)(3)) \mathrm{I} = (10a+3b) \mathrm{I} = \left[\begin{array}{cc} 10a+3b & 0 \\ 0 & 10a+3b \end{array}\right] \) \( \mathrm{AA}^{\mathrm{T}} = \left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right] \left[\begin{array}{cc}5 a & 3 \\ -b & 2\end{array}\right] = \left[\begin{array}{cc} 25a^2+b^2 & 15a-2b \\ 15a-2b & 9+4 \end{array}\right] = \left[\begin{array}{cc} 25a^2+b^2 & 15a-2b \\ 15a-2b & 13 \end{array}\right] \)
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