Nilpotent Matrix – Definition and Example

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

$$\implies$$ $$A^2$$ = $$\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}$$ $$\times$$ $$\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}$$

= $$\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}$$.

Now, we have to find $$A^3$$ = $$A^2$$.A

$$\implies$$ $$A^3$$ = $$\begin{bmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{bmatrix}$$ $$\times$$ $$\begin{bmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{bmatrix}$$

= $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ = O

$$\therefore$$ $$A^3$$ = O i.e. $$A^k$$ = O

Here k =3

Hence A is a nilpotent matrix of order 3.