JEE Mains · Maths · STD 12 - 3 and 4 . metrices and determinant
Let \(A\) be a \(3 \times 3\) matrix such that \(A^2 -5A+ 7I = 0\). Statement \(-I\) : \({A^{ - 1}} = \frac{1}{7}\left( {5I - A} \right).\) Statement \(II\) : the polynomial \(A^3 - 2A^2 - 3A + I\) can be reduced to \(5\, (A - 4I)\).
- A Both the statements are true
- B Both the statements are false
- C Statement \(-I\) is true, but Statement \(-II\) is fulse
- D Statement \(I\) is false, but Statement \(-II\) is true
Answer & Solution
Correct Answer
(A) Both the statements are true
Step-by-step Solution
Detailed explanation
\({A^2} - 5A = - 7I\) \(AA{A^{ - 1}} - 5A{A^{ - 1}} = - 7I{A^{ - 1}}\) \(AI - 5I = - 7{A^{ - 1}}\) \(A - 5I = - 7{A^{ - 1}}\) \({A^{ - 1}} = \frac{1}{7}\left( {5I - A} \right)\) \({A^3} - 2{A^2} - 3A + I\) \( = A\left( {5I - 7I} \right) - 2{A^2} - 3A + I\)…
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