Matrix Polynomial – Definition and Example

Here you will learn what is the matrix polynomial definition with examples.

Let’s begin –

Matrix Polynomial

Definition : Let f(x) = \(a_0x^n\) + \(a_1x^{n – 1}\) + \(a_2x^{n – 2}\) + ……. + \(a_{n – 1}x\) + \(a_n\) be a polynomial and let A be a square matrix of order n. Then,

f(A) = \(a_0A^n\) + \(a_1A^{n – 1}\) + \(a_2A^{n – 2}\) + ….  \(a_{n – 1}A\) + \(a_nI^n\)

is called a matrix polynomial.

For example, if f(x) = \(x^2\) – 3x + 2 is a polynomial and A is a square matrix, then f(A) = \(A^2\) – 3A + 2I is a matrix polynomial.

Also Read : Different Types of Matrices – Definitions and Examples

Example : Let f(x) =  \(x^2\) – 4x + 7. Find f(A), if A = \(\begin{bmatrix} 2 & 3 \\ -1 &  2 \end{bmatrix}\).

Solution : We have, f(x) =  \(x^2\) – 4x + 7

\(\therefore\)   f(A) = \(A^2 – 4A + 7I_2\)

Now,  \(A^2\) = \(\begin{bmatrix} 2 & 3 \\ -1 &  2 \end{bmatrix}\)\(\begin{bmatrix} 2 & 3 \\ -1 &  2 \end{bmatrix}\)

= \(\begin{bmatrix} 4 – 3 & 6 + 6 \\ -2 – 2 &  -3 + 4 \end{bmatrix}\) = \(\begin{bmatrix} 1 & 12 \\ -4 &  1 \end{bmatrix}\)

-4A = \(\begin{bmatrix} -8 & -12 \\ 4 &  -8 \end{bmatrix}\)

and, \(7I_2\) = \(\begin{bmatrix} 7 & 0 \\ 0 &  7 \end{bmatrix}\)

\(\therefore\)   f(A) = \(A^2 – 4A + 7I_2\)

\(\implies\)  f(A) = \(\begin{bmatrix} 1 & 12 \\ -4 &  1 \end{bmatrix}\) + \(\begin{bmatrix} -8 & -12 \\ 4 &  -8 \end{bmatrix}\) + \(\begin{bmatrix} 7 & 0 \\ 0 &  7 \end{bmatrix}\)

\(\implies\)  f(A) = \(\begin{bmatrix} 0 & 0 \\ 0 &  0 \end{bmatrix}\)

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