How to Prove a Matrix is Idempotent?

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To prove that the matrix A = I - X(X'X)^{-1}X' is idempotent, one can either show that the matrix X itself is idempotent before working with the entire expression or directly manipulate the entire expression. Both methods are valid and depend on careful application of matrix multiplication properties, particularly the associative property. Understanding the context of this matrix in statistics, particularly in multiple linear regression, can also aid in grasping its properties. Mastery of these techniques will facilitate solving similar matrix problems effectively.
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A=I - X(X'X)^-1X'

Show that Matrix A is idempotent.

I'm new to matrices and am having trouble proving this. Could anyone give me a hand as far as how to get started on solving this problem and possibly some tips for how to do problems like it. Thanks.
 
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I'm guessing this question comes from statistics (multiple linear regression?), because the matrix

<br /> I - X(X&#039;X)^{-1}X&#039;<br />

generates the residuals in that topic.

Two ways to go - neither is any better than the other
1) Show the "X" portion itself is idempotent, then work with the entire thing
2) Move directly to working with the entire expression

Either approach requires judicious use of parentheses and the associative property of matrix multiplication.
 

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