Calculators / Linear Algebra / Matrix Power Calculator

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. Make sure A is square and choose an exponent n.
  2. Write Aⁿ as repeated multiplication of A.
  3. Multiply step-by-step to reach Aⁿ.

Matrix power formula

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

Example 1: A (2×2), n = 3 → A^3

Step 1 - Make sure A is square and choose an exponent n.

In this problem: A is 2×2 and n = 3.

A =
1
2
0
1
, n = 3

Step 2 - Write Aⁿ as repeated multiplication of A.

In this problem: Write A3 as repeated multiplication.

1
2
0
1
×
1
2
0
1
×
1
2
0
1

Step 3 - Multiply step-by-step to reach Aⁿ.

In this problem: Multiply A1 by A to get A2.

1
2
0
1
×
1
2
0
1

Step 4 - Compute the entries of A<sup>2</sup>.

In this problem: Each entry is a row-by-column dot product.

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

Step 5 - Multiply step-by-step to reach Aⁿ.

In this problem: Multiply A2 by A to get A3.

1
4
0
1
×
1
2
0
1

Step 6 - Compute the entries of A<sup>3</sup>.

In this problem: Each entry is a row-by-column dot product.

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: A (3×3), n = 2 → A^2

Step 1 - Make sure A is square and choose an exponent n.

In this problem: A is 3×3 and n = 2.

A =
2
0
-1
1
3
2
0
1
1
, n = 2

Step 2 - Write Aⁿ as repeated multiplication of A.

In this problem: Write A2 as repeated multiplication.

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

Step 3 - Multiply step-by-step to reach Aⁿ.

In this problem: Multiply A1 by A to get A2.

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

Step 4 - Compute the entries of A<sup>2</sup>.

In this problem: Each entry is a row-by-column dot product.

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
n