Suppose A is any \(3 \times 3\) non-singular matrix and \((A - 3I)(A - 5I) = O\) where \(I = I_3\) and \(O = O_3\). If \(\alpha A + \beta A^{-1} = 4I\), then \(\alpha + \beta\) is equal to :
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- \(8\)
- \(7\)
- \(13\)
- \(12\)
The Correct Option is A
Solution and Explanation
Multiply by \(A^{-1}\) (A is non-singular): \(A - 8I + 15A^{-1} = O\), so \(A + 15A^{-1} = 8I\).
Divide by 2: \(\tfrac12 A + \tfrac{15}{2}A^{-1} = 4I\).
Thus \(\alpha = \tfrac12,\ \beta = \tfrac{15}{2}\) and \(\alpha+\beta = 8\).
Answer: (1) \(8\)
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