On this post we explain how to calculate the power of a matrix. Here you will find examples and exercises solved step by step of matrix powers that will help you understand it perfectly. You will also learn what the nth power of a matrix is and how to find it.

Table of Contents

## How to find the power of a matrix?

To find the power of a matrix, multiply the matrix by itself as many times as the exponent indicates.

Therefore, to calculate the power of a matrix, you must first know how to multiply matrices. Otherwise you will not be able to calculate the power of a matrix. Then, before reading any further, we recommend that you first see the matrix multiplication algorithm.

### Example of the power of a matrix

Let A be a 2×2 square matrix, the 4th power of matrix A is calculated as follows:

There is an important property of matrix power that you must know: you can only calculate the power of a matrix when it is a square matrix.

The power of a matrix can also be calculated using using eigenvalues, that is, by diagonalizing the matrix. However, you have to know how to do a matrix diagonalization. In this link you will find how to diagonalize a matrix and also how to calculate the power of a matrix by diagonalization.

## What is the nth power of a matrix?

The **nth power of a matrix** is an expression that allows us to calculate any power of a matrix easily.

Many times powers of matrices follow a pattern. Therefore, if we find the sequence that the powers of a matrix follow, we can calculate any power without having to do all the multiplications.

This means that we can find a **formula that gives us the nth power of a matrix** without having to calculate all the powers.

The formula of some matrix powers can be found as follows:

- The
**parity of the exponent**. It may be that the even powers are one way and odd powers another. **Variation in signs.**For example, it could be that the elements of the even powers are positive and the elements of the odd powers negative, or vice versa.**Repetition:**whether the same matrix is repeated every a certain number of powers or not.- There is a
**mathematical relationship**between the exponent and the elements of the matrix.

### Example of the nth power of a matrix

- Let A be a the following matrix of order 2, find A
^{n}and A^{100}:

First we are going to calculate several powers of matrix A to try to guess the pattern that the powers follow. So we calculate the five first powers of the matrix:

When calculating up to A^{5}, we see that the powers of matrix A follow a pattern: with each increase in power the result is multiplied by 2. Therefore, **all the elements of the matrices are powers of 2:**

So we can deduce by induction that the formula for the nth power of matrix A is as follows:

And from this formula we can calculate matrix A raised to 100:

## Practice matrix power

### Problem 1

Given the following 2×2 dimension matrix:

Raise the matrix to the fourth power.

**See solution**

To calculate the power of a matrix, we have to multiply the matrix one by one. Therefore, we first calculate the square of matrix A:

Now we calculate the cube of matrix A:

And finally we determine A^{4}:

### Problem 2

Given the following matrix of order 2:

Calculate:

**See solution**

A^{35} is a power too large to calculate by hand, therefore the powers of the matrix must follow a pattern. So we’re going to calculate up to A^{5} to try to figure out the sequence:

Now we can see the pattern that the powers follow: at each power all numbers remain the same, except for the element in the second column of the second row, which is multiplied by 3. Therefore, **all numbers always remain the same and the last element is a power of 3:**

Thus, the formula for the nth power of matrix A is:

And from this formula we can calculate A^{35}:

### Problem 3

Given the following 3×3 dimension matrix:

Calculate:

**See solution**

A^{100} is a power too large to calculate by hand, which means that the powers must follow a pattern. So we’re going to calculate up to A^{5} to try to figure out the sequence:

Thus, at each power all numbers remain the same except for the fractions, which **increase by one unit in the numerator:**

So the formula that defines the nth power of matrix A is:

And from this formula we can find A^{100}:

### Problem 4

Given the following matrix of size 2×2:

Find:

**See solution**

A^{201} is a power too large to calculate by hand, therefore the powers of the matrix must follow a pattern. In this case, we must calculate up to A^{8} in order to find out their sequence:

With these calculations we can see that every 4 powers we obtain the identity matrix. That is, powers A^{4}, A^{8}, A^{12}, A^{16}, … will result in the identity matrix. So to calculate A^{201} we have to decompose 201 into multiples of 4:

Thus, A^{201} will be 50 times A^{4} and once A^{1}:

Since we know that A^{4} is the identity matrix:

Besides, the identity matrix raised to any number gives the identity matrix. So:

And finally, any matrix multiplied by the identity matrix results in the same matrix. Thus:

So A^{201} is equal to A:

### Problem 5

Given the following square matrix of order 3:

Calculate:

**See solution**

Obviously, A^{62} is a calculation too large a to do by hand, so the powers of the matrix have to follow a pattern. In this case we must calculate up to A^{6} in order to find out the sequence that they follow:

With these calculations we can see that every 3 powers we obtain the identity matrix. In other words, powers A^{3}, A^{6}, A^{9}, A^{12}, … result in the identity matrix. So to calculate A^{62} we have to decompose 62 into multiples of 3:

Thus, A^{62} is 20 times A^{3} and once A^{2}:

Since we know that A^{3} is the identity matrix:

Besides, the identity matrix raised to any number gives the identity matrix. So:

Finally, any matrix multiplied by the identity matrix results in the same matrix. So:

So A^{62} will be equal to A^{2}:

TallulahThe explanation is very clear, thanks a lot!

[email protected]Glad to be of help!