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Q.
Let $M$ be a $3 \times 3$ matrix with real entries. If $M^{T} M=I_{3}$, where $I_{3}$ is a $3 \times 3$ identity matrix, then find the value of det. $\left(M^{2}-I_{3}\right)$.
Matrices
Solution:
det. $\left(M M-M^{\prime} M\right)=$ det. $\left(M-M^{\prime}\right) \cdot$ det. $M$
$\because M$ is $3^{\text {rd }}$ order matrix
$\Rightarrow M-M^{\prime}$ is skew-symmetric matrix
$\Rightarrow det\left(M-M^{\prime}\right)=0$