# 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.

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