Let M be ann x n matrix. (a) (10 points) M is called nilpotent if there...

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Let M be ann x n matrix. (a) (10 points) M is called nilpotent if there exists k e N such that Mk is the zero matrix. Prove that det M = 0 if M is nilpotent. (b) (10 points) M is called skew-symmetric if its transpose is equal to -M. Assume that M is skew-symmetric and invertible. Prove that n is even. Let M be ann x n matrix. (a) (10 points) M is called nilpotent if there exists k e N such that Mk is the zero matrix. Prove that det M = 0 if M is nilpotent. (b) (10 points) M is called skew-symmetric if its transpose is equal to -M. Assume that M is skew-symmetric and invertible. Prove that n is even.