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An idempotent matrix is one which, when multiplied by itself, doesn’t change. If a matrix A is idempotent, A2 = A.
Examples of Idempotent Matrix
The simplest examples of n x n idempotent matrices are the identity matrix In, and the null matrix (where every entry on the matrix is 0).
Nontrivial examples of 2 x 2 matrices are relatively easy to come up with (Need help? Check out our tutoring page!). Since A2 = A, we know that for a matrix 
,
a = a2 + bc
b = ab + bd, so b – ab – bd = 0 and b(1 – a – d) = 0 and either b = 0 or d = 1 – a
c = ca + cd, so c – ca – cd = 0 and c(1 – a – d) = 0 and either c = 0 or d = 1 – a
d = bc + d2.
To come up with your own idempotent matrix, start by choosing any value of a. For an example, let’s choose 3. Make d = 1 – a, or -2. Now -2 = bc + 4, by our a last equation above, so -6 = bc. As a quick check, see that this fits our first equation, a = 3 = a2 + bc = 9 – 6.
We can choose any values of b and c that fit bc = -6, so to make it easy on ourselves and stick to whole integers; b = -2 and c = 3. So the matrix
is idempotent.
You can come up with a similar set of algebra rules for finding idempotent matrices in any size n x n ; the larger the matrix gets, the stickier the algebra may become, but the method remains the same.
Properties of Idempotent Matrices
Except for the identity matrix (I), every idempotent matrix is singular. What this means is that it is a square matrix, whose determinant is 0.
Since
[I – M] [I – M] = I – M – M + M2 = I – M – M + M = I – M,
the identity matrix minus any other idempotent matrix is also an idempotent matrix.
The Idempotent Matrix in Statistics
Idempotent matrices are important in regression analysis and the theory of linear statistical models, especially as they relate to the analysis of variance and the theory of least squares.
References
Harville D.A. (1997) Idempotent Matrices. In: Matrix Algebra From a Statistician’s Perspective.