Non-Negative Integer Powers of Matrices

Just like with scalars, non-negative integer powers of matrices are defined in terms of multiplication.

That is

A^n = \underbrace{AA \cdots A}_{n \text{ times}}.

Just like any scalar $a$ to the power of $0$ is defined to be the multiplicative identity ( $1$ ), that is also the case with matrices. A matrix to the $0$ th power always equals the identity matrix of that size.

That is

A_{n \times n}^0 = I_n.

It is important to note that powers of non-square matrices are not defined, since conformability would not be satisfied. If you wanted to multiply a matrix of size $A_{m \times n}$ with itself, this would only be defined when $m = n$ . Therefore, $A$ must be a square matrix.

Example

To illustrate, here is a concrete numerical example. Take the matrix

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}.

We compute

\begin{align*} AA &= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \\ &= \begin{bmatrix} 1 \cdot 1 + 2 \cdot 3 & 1 \cdot 2 + 2 \cdot 4 \\ 3 \cdot 1 + 4 \cdot 3 & 3 \cdot 2 + 4 \cdot 4 \end{bmatrix} \\ &= \begin{bmatrix} 7 & 10 \\ 15 & 22 \end{bmatrix}. \end{align*}

Properties of Powers of Matrices

The common rules for powers and exponents known from scalar algebra also hold for matrix powers.

Grouping of Exponents

Because matrix multiplication is associative, you can group the powers however you want:

A^4 = AAAA = A^2 A^2 = A^3 A = A A^3.

Multiplication of Powers

Just like with scalars, multiplying to powers with the same basis is the same as adding the exponents:

A^r A^s = A^{r + s}.

Powers of Powers

Similarly, the power of a power of a matrix is the same as multiplying the exponents:

(A^r)^s = A^{rs}

An Important Warning

Keep in mind that, since matrix multiplication is not commutative, when expanding

\begin{align*} (A + B)^2 &= (A + B)(A + B) \\ &= (A + B)A + (A + B)B \\ &= A^2 + BA + AB + B^2, \end{align*}

this is not the same as

A^2 + 2AB + B^2.

In fact, generally speaking

(A + B)^2 \ne A^2 + 2AB + B^2,

because in general $AB + BA \ne 2AB$ , since $AB \ne BA$ .

A Final Remark

Fractional and negative exponents can also be defined for matrices. However, with the content covered so far, we aren’t yet equipped to explore them in detail.