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
- Check that Matrix A is square and choose the exponent n.
- Write Aⁿ as Matrix A multiplied by itself n times.
- Multiply step by step until the final power is reached.
Formula:
Example 1: Take the values below.
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^{3}\)
Step 1 - Check that Matrix A is square and choose the exponent n.
In this problem: Matrix A is \(2 \times 2\), so it is square. We are finding \(A^{3}\).
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^{3}\)
Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.
In this problem: \(A^{3}\) means Matrix A appears \(3\) times.
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 3 - Set up A^2.
In this problem: Start by multiplying A by A.
\(A^{2} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 4 - Compute A^2.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 1 \cdot 1 + 2 \cdot 0 & 1 \cdot 2 + 2 \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}\)
Step 5 - Set up A^3.
In this problem: Use the result from \(A^{2}\), then multiply by A.
\(A^{3} = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 6 - Compute A^3.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 1 \cdot 1 + 4 \cdot 0 & 1 \cdot 2 + 4 \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\)
Final answer:
\(A^{3} = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\)
Example 2: Take the values below.
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}^{2}\)
Step 1 - Check that Matrix A is square and choose the exponent n.
In this problem: Matrix A is \(3 \times 3\), so it is square. We are finding \(A^{2}\).
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}^{2}\)
Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.
In this problem: \(A^{2}\) means Matrix A appears \(2\) times.
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}\)
Step 3 - Set up A^2.
In this problem: Start by multiplying A by A.
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}\)
Step 4 - Compute A^2.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 2 \cdot 2 + 0 \cdot 1 + (-1) \cdot 0 & 2 \cdot 0 + 0 \cdot 3 + (-1) \cdot 1 & 2 \cdot (-1) + 0 \cdot 2 + (-1) \cdot 1 \\ 1 \cdot 2 + 3 \cdot 1 + 2 \cdot 0 & 1 \cdot 0 + 3 \cdot 3 + 2 \cdot 1 & 1 \cdot (-1) + 3 \cdot 2 + 2 \cdot 1 \\ 0 \cdot 2 + 1 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot 3 + 1 \cdot 1 & 0 \cdot (-1) + 1 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 4 & -1 & -3 \\ 5 & 11 & 7 \\ 1 & 4 & 3 \end{pmatrix}\)
Final answer:
\(A^{2} = \begin{pmatrix} 4 & -1 & -3 \\ 5 & 11 & 7 \\ 1 & 4 & 3 \end{pmatrix}\)
See Example 1
Hide Example 1
Problem
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^{3}\)
Approach
Step-by-step method
- Check that Matrix A is square and choose the exponent n.
- Write Aⁿ as Matrix A multiplied by itself n times.
- Multiply step by step until the final power is reached.
Step 1
Step 1 - Check that Matrix A is square and choose the exponent n.
In this problem: Matrix A is \(2 \times 2\), so it is square. We are finding \(A^{3}\).
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}^{3}\)
Step 2
Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.
In this problem: \(A^{3}\) means Matrix A appears \(3\) times.
\(A^{3} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 3
Step 3 - Set up A^2.
In this problem: Start by multiplying A by A.
\(A^{2} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 4
Step 4 - Compute A^2.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 1 \cdot 1 + 2 \cdot 0 & 1 \cdot 2 + 2 \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}\)
Step 5
Step 5 - Set up A^3.
In this problem: Use the result from \(A^{2}\), then multiply by A.
\(A^{3} = \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix} \times \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
Step 6
Step 6 - Compute A^3.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 1 \cdot 1 + 4 \cdot 0 & 1 \cdot 2 + 4 \cdot 1 \\ 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\)
Final Answer
\(A^{3} = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\)
See Example 2
Hide Example 2
Problem
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}^{2}\)
Approach
Step-by-step method
- Check that Matrix A is square and choose the exponent n.
- Write Aⁿ as Matrix A multiplied by itself n times.
- Multiply step by step until the final power is reached.
Step 1
Step 1 - Check that Matrix A is square and choose the exponent n.
In this problem: Matrix A is \(3 \times 3\), so it is square. We are finding \(A^{2}\).
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}^{2}\)
Step 2
Step 2 - Write Aⁿ as Matrix A multiplied by itself n times.
In this problem: \(A^{2}\) means Matrix A appears \(2\) times.
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}\)
Step 3
Step 3 - Set up A^2.
In this problem: Start by multiplying A by A.
\(A^{2} = \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix} \times \begin{pmatrix} 2 & 0 & -1 \\ 1 & 3 & 2 \\ 0 & 1 & 1 \end{pmatrix}\)
Step 4
Step 4 - Compute A^2.
In this problem: Multiply each row by each column, then add the products.
\(\begin{pmatrix} 2 \cdot 2 + 0 \cdot 1 + (-1) \cdot 0 & 2 \cdot 0 + 0 \cdot 3 + (-1) \cdot 1 & 2 \cdot (-1) + 0 \cdot 2 + (-1) \cdot 1 \\ 1 \cdot 2 + 3 \cdot 1 + 2 \cdot 0 & 1 \cdot 0 + 3 \cdot 3 + 2 \cdot 1 & 1 \cdot (-1) + 3 \cdot 2 + 2 \cdot 1 \\ 0 \cdot 2 + 1 \cdot 1 + 1 \cdot 0 & 0 \cdot 0 + 1 \cdot 3 + 1 \cdot 1 & 0 \cdot (-1) + 1 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 4 & -1 & -3 \\ 5 & 11 & 7 \\ 1 & 4 & 3 \end{pmatrix}\)
Final Answer
\(A^{2} = \begin{pmatrix} 4 & -1 & -3 \\ 5 & 11 & 7 \\ 1 & 4 & 3 \end{pmatrix}\)