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.

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