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A singular matrix is a matrix that is not invertible. If a matrix is not invertible, then:

• The determinant of the matrix is 0.

• Any matrix multiplied by that matrix doesn't give the identity matrix.

There are a lot of examples in which a singular matrix is an idempotent matrix. For instance:

M =

[1 1]

[0 0]

Take the product of two M's to get the same M, the given!

M x M = M

So yes, SOME singular matrices are idempotent matrices! How? Let's take a 2 by 2 identity matrix for instance.

I =

[1 0]

[0 1]

I x I = I obviously.

Then, that nonsingular matrix is also idempotent!

Hope this helps!

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Q: Is a singular matrix an indempotent matrix?
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