Matrix Power Calculator

Published on: October 19 2025

This Matrix Power Calculator helps you raise a square matrix to a power and shows each step clearly. It works by multiplying the matrix by itself the required number of times to compute An. This makes it useful for checking answers, understanding how matrix powers work, and practising linear algebra step by step.

Step-by-step method

  1. Check that Matrix A is square and choose the exponent n.
  2. Write Aⁿ as Matrix A multiplied by itself n times.
  3. Multiply step by step until the final power is reached.

Formula:

An = A × A × ... × A
where A is square and appears n times

Example 1: Take the matrix and exponent below.

A3 =
1
2
0
1
3

Step 1 - Check that Matrix A is square and choose the exponent n.

In this problem: Matrix A is 2×2, so it is square. We are finding A3.

A3 =
1
2
0
1
3

Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.

In this problem: A3 means Matrix A appears 3 times.

1
2
0
1
×
1
2
0
1
×
1
2
0
1

Step 3 - Set up A<sup>2</sup>.

In this problem: Start by multiplying A by A.

A2 =
1
2
0
1
×
1
2
0
1

Step 4 - Compute A<sup>2</sup>.

In this problem: Each entry is found using row-by-column multiplication.

A2 =
1×1 + 2×0
1×2 + 2×1
0×1 + 1×0
0×2 + 1×1
=
1
4
0
1

Step 5 - Set up A<sup>3</sup>.

In this problem: Use the result from A2, then multiply by A.

A3 =
1
4
0
1
×
1
2
0
1

Step 6 - Compute A<sup>3</sup>.

In this problem: Each entry is found using row-by-column multiplication.

A3 =
1×1 + 4×0
1×2 + 4×1
0×1 + 1×0
0×2 + 1×1
=
1
6
0
1

Final answer:

1
6
0
1

Example 2: Take the matrix and exponent below.

A2 =
2
0
-1
1
3
2
0
1
1
2

Step 1 - Check that Matrix A is square and choose the exponent n.

In this problem: Matrix A is 3×3, so it is square. We are finding A2.

A2 =
2
0
-1
1
3
2
0
1
1
2

Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.

In this problem: A2 means Matrix A appears 2 times.

2
0
-1
1
3
2
0
1
1
×
2
0
-1
1
3
2
0
1
1

Step 3 - Set up A<sup>2</sup>.

In this problem: Start by multiplying A by A.

A2 =
2
0
-1
1
3
2
0
1
1
×
2
0
-1
1
3
2
0
1
1

Step 4 - Compute A<sup>2</sup>.

In this problem: Each entry is found using row-by-column multiplication.

A2 =
2×2 + 0×1 + (-1)×0
2×0 + 0×3 + (-1)×1
2×(-1) + 0×2 + (-1)×1
1×2 + 3×1 + 2×0
1×0 + 3×3 + 2×1
1×(-1) + 3×2 + 2×1
0×2 + 1×1 + 1×0
0×0 + 1×3 + 1×1
0×(-1) + 1×2 + 1×1
=
4
-1
-3
5
11
7
1
4
3

Final answer:

4
-1
-3
5
11
7
1
4
3
Matrix A Setup
Exponent
Matrix A
2