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Show that a given matrix is symmetric and idempotent

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Let X be a txk matrix whose columns are linearly independent. Let M=I-X(X'X)^(-1)X'. Show that M is symmetric and idempotent.

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The solution walks the student through the problem of proving that a given matrix is symmetric and idempotent. A step-by-step solution is provided for the student.

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M is symmetric if M'=M. So let's check to see if this true.

M'=[I-X(X'X)^(-1)X']'
To transpose the second part of this matrix (after the minus side), all we do is take each element in reverse order and transpose it. Let's do ...

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