Idempotent Matrix – Definition and Example

Here you will learn what is idempotent matrix with examples.

Let’s begin –

Idempotent Matrix

A square matrix is idempotent matrix provided \(A^2\) = A.

For this matrix note the following :

(i) \(A^n\) = A \(\forall\) n \(\ge\) 2, n \(\in\) N.

(ii) The determinant value of this matrix is either 1 or 0.

Example : Show that the matrix A = \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\) is idempotent.

Solution : We have,

A = \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\)

Now, \(A^2\) = A.A

\(\implies\) A = \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\) \(\times\) \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\)

= \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\) = A

Hence, matrix A is idempotent.

Example : Find the determinant of above matrix A = \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\)

Solution : We have,

A = \(\begin{bmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{bmatrix}\)

Now, | A | = \(\begin{vmatrix} 2 & -2 &  -4 \\  -1 & 3 & 4 \\  1 & -2 & -3 \end{vmatrix}\)

\(\implies\) | A | = 2 \(\begin{vmatrix} 3 & 4 \\ -2 & -3 \end{vmatrix}\) – (-2) \(\begin{vmatrix} -1 & 4 \\  1 & -3 \end{vmatrix}\) + (-4) \(\begin{vmatrix}  -1 & 3 \\  1 & -2 \end{vmatrix}\)

\(\implies\) | A | = 2 (-9 + 8) + 2 (3 – 4) – 4 ( 2 – 3) 

= 2(-1) + 2(-1) – 4(-1)

= -2 –  2 + 4 = 0

Hence, determinant of matrix A is 0.

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