Here you will learn what is nilpotent matrix with examples.
Let’s begin –
Nilpotent Matrix
A square matrix of the order ‘n’ is said to be a nilpotent matrix of order m, m \(\in\) N
if \(A^m\) = O & \(A^{m-1}\) \(\ne\) O.
Example : Show that A = \(\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\) is a nilpotent matrix of order 3.
Solution : We have given the matrix A,
A = \(\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\)
Now first we find, \(A^2\) = A.A
\(\implies\) \(A^2\) = \(\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\) \(\times\) \(\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\)
\(\implies\) \(A^2\) = \(\begin{bmatrix} 1+5-6 & 1+2-3 & 3+6-9 \\ 5+10-12 & 5+4-6 & 15+12-18 \\ -2-5+6 & -2-2+3 & -6-6+9 \end{bmatrix}\)
= \(\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}\).
Now, we have to find \(A^3\) = \(A^2\).A
\(\implies\) \(A^3\) = \(\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}\) \(\times\) \(\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}\)
\(\implies\) \(A^3\) = \(\begin{bmatrix} 0+0+0 & 0+0+0 & 0+0+0 \\ 3+15-18 & 3+6-9 & 9+18-37 \\ -1-5+6 & -1-2+3 & -3-6+9 \end{bmatrix}\)
= \(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\) = O
\(\therefore\) \(A^3\) = O i.e. \(A^k\) = O
Here k =3
Hence A is a nilpotent matrix of order 3.