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.
