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3 and 4 .Determinants and Matrices
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If $A$ is nilpotent matrix of index $2$, then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)
A
$A^{51}$
B
$I_2$
C
Null matrix
D
$A$
Solution
$A^{2}=O \Rightarrow A^{3}=O, A^{4}=O$
$ \Rightarrow \mathrm{A}^{51}= \mathrm{O}$
$(\mathrm{I}+\mathrm{A})^{2} =\mathrm{A}^{2}+2 \mathrm{A}+\mathrm{I}=2 \mathrm{A}+\mathrm{I} $
$(\mathrm{I}+\mathrm{A})^{3} =(\mathrm{I}+\mathrm{A})(2 \mathrm{A}+\mathrm{I}) $
$=2 \mathrm{A}^{2}+3 \mathrm{A}+\mathrm{I}$
$=3 \mathrm{A}+\mathrm{I}$
similarly $(\mathrm{I}+\mathrm{A})^{51}=51 \mathrm{A}+\mathrm{I}$
$\mathrm{A}(\mathrm{I}+\mathrm{A})^{51}=51 \mathrm{A}^{2}+\mathrm{A}=\mathrm{A}$
Standard 12
Mathematics
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