Question

If A = $\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$, then which one of the following statements is not correct ?

• A. $A^4 - I = A^2 +I$
• B. $A^3 - I = A(A - I)$
• C. $A^2 + I = A(A^2 - I)$
• D. $A^3 + I = A(A - I)$

Answer 1

Answer: (C)

Given that
$A = \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}$
$A^{2} = \begin{bmatrix}-1&0\\ 0&-1\end{bmatrix} \Rightarrow A^{2} = -I$
$A^{3} = \begin{bmatrix}0&1\\ -1&0\end{bmatrix}$
$A^{4} = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} = I$
$A^{2} +I = A^{3}-A$
$-I + I = A^{3}-A$
$A^{3} \ne A$