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3 and 4 .Determinants and Matrices
hard
If $A\, = \,\left[ {\begin{array}{*{20}{c}}
0&{ - 1}\\
1&0
\end{array}} \right],$ then which one of the following statements is not correct?
A
$A^2 + I = A(A^2 - I)$
B
$A^4 - I = A^2 + I$
C
$A^3 + I = A(A^3 - I)$
D
$A^3 - I = A(A- I)$
(JEE MAIN-2015)
Solution
Given that
$A = \left[ {\begin{array}{*{20}{c}}
0&{ – 1}\\
1&0
\end{array}} \right]$
${A^2} = \left[ {\begin{array}{*{20}{c}}
{ – 1}&0\\
0&{ – 1}
\end{array}} \right] \Rightarrow {A^2} = – I$
${A^3} = \left[ {\begin{array}{*{20}{c}}
0&1\\
{ – 1}&0
\end{array}} \right]$
${A^4} = \left[ {\begin{array}{*{20}{c}}
1&0\\
0&1
\end{array}} \right] = I$
${A^2} + I = {A^3} – A$
$ – I + I = {A^3} – A$
${A^3} \ne A$
Standard 12
Mathematics
