If \(A = \begin{bmatrix} 0\;\;-1\\1\;\;0 \end{bmatrix}\), then which one of the following statements is not correct?
Show Hint
- \(A^3 + I = A(A^3 - I)\)
- \(A^4 - I = A^2 + I\)
- \(A^2 + I = A(A^2 - I)\)
- \(A^3 - I = A(A - I)\)
The Correct Option is C
Solution and Explanation
(1) LHS \(= -A + I\); RHS \(= A(-A - I) = -A^2 - A = I - A\). Equal, so correct.
(2) LHS \(= I - I = 0\); RHS \(= -I + I = 0\). Equal, so correct.
(3) LHS \(= -I + I = 0\); RHS \(= A(-I - I) = -2A \neq 0\). Not equal, so this statement is not correct.
(4) LHS \(= -A - I\); RHS \(= A^2 - A = -I - A\). Equal, so correct.
Answer: (3) \(A^2 + I = A(A^2 - I)\)
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