Question

If $A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right],$ then which one of the following statements is not correct ?

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

Answer 1

The correct option is C $A^2+I=A(A^2–I)$

$A=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$
$A^2=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

$\Rightarrow A^2=-I$
$\Rightarrow A^3=A\left(A^2\right)=A(-I)=-A\Rightarrow A^4=(A^2{)}^2=(-I{)}^2=I$

Now, check options one by one
$\to A^3–I=A(A–I)$
$\text{L.H.S.}=A^3–I=-A-I$
$\text{R.H.S.}=A(A–I)=A^2-AI=-A-I$

$\to A^3+I=A(A^3–I)$
$\text{L.H.S.}=A^3+I=-A+I$
$\text{R.H.S.}=A(A^3–I)=A^4-AI=I-A$

$\to A^2+I=A(A^2–I)$
$\text{L.H.S.}=A^2+I=-I+I=O$
$\text{R.H.S.}=A(A^2–I)=A(-I-I)=-2AI=-2A$
$\text{L.H.S.}\ne \text{R.H.S.}$

$\to A^4-I=A^2+I$
$\text{L.H.S.}=A^4-I=I-I=O$
$\text{R.H.S.}=A^2+I=-I+I=O$