Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N 2 and M 2 = N 4 , then

M 2 = N 4 ⇒ M 2 - N 4 = 0 ⇒ M - N 2 M + N 2 = 0 (As M, N commute.) Also, M ≠ N 2 , D e t M - N 2 M + N 2 = 0 As M - N 2 is not null ⇒ D e t M + N 2 = 0 Also D e t M 2 + M N 2 = D e t M D e t M + N 2 = 0 ⇒ There exist non - null U such that M 2 + M N 2 U = 0

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MathematicsMatrixGeneralMedium2 minPYQ_2014

MathematicsMediummultiple choice

Let M and N be two $3 \times 3$ matrices such that MN = NM. Further, if $M \neq N^{2}$ and $M^{2} = N^{4}$ , then

Options:(select one or more)

Answer:

A, B

Solution:

$M^{2} = N^{4} \Rightarrow M^{2} - N^{4} = 0 \Rightarrow (M - N^{2}) (M + N^{2}) = 0$ (As M, N commute.)
Also, $M \neq N^{2}, D e t ((M - N^{2}) (M + N^{2})) = 0$
As $M - N^{2}$ is not null $\Rightarrow D e t (M + N^{2}) = 0$
Also $D e t (M^{2} + M N^{2}) = (D e t M) (D e t (M + N^{2})) = 0$
$\Rightarrow$ There exist non - null U such that $(M^{2} + M N^{2}) U = 0$

Stream:JEE_ADVSubject:MathematicsTopic:MatrixSubtopic:General

⏱ 2mℹ️ Source: PYQ_2014

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