## Introduction to Matrix Power Calculator:

A matrix power calculator is an online tool that helps you **find an integer**'s given square matrix-specific power in a few seconds. Our tool allows you to evaluate the exponential power of a matrix A, which may be 1, 2, 3, 4,..., n in n x m order.

It is a powerful tool for students, educators, and professionals as it can quickly perform matrix exponentiation power operations.

## What is Matrix Power?

Matrix power is a process used to calculate the **square matrix A** integer exponent. If A is a square matrix (the number of rows is equal to the number of columns), it means A^{k} represents the matrix A that is multiplied by itself k times.

Matrix power is a fundamental operation in linear algebra because it has various applications in different fields except mathematics, physics, graphics, computers, etc. It can be expressed as:

$$ A^k \;=\; A \times A \times … \times A \; (k\; times) $$

Here, k is the integer, where k = 0, 1, 2, 3, 4,..., and A is the square matrix.

## How to Calculate Matrix Power?

For evaluating the power of a matrix, you need to **multiply the matrix** by itself repeatedly until the given integer k number is not achieved.

Three methods, direct, exponential squaring, and diagonalization are used to find the matrix power problem. Let's see the working process of these methods one by one.

### Direct Method:

In this method, you **directly multiply the matrix** by itself till the required number of k times.

For a matrix A and power k:

$$ A^k \;=\; A \times A \times ⋯ \times A\; (k\; times) $$

$$ A^k \;=\; A $$

### Exponentiation by Squaring

This method reduces the number of multiplications that are needed with the help of some rules. It is particularly used for large powers. If k=0, the solution becomes identity matrix I, or If k=1, you get **matrix A** itself. If k is even, then compute A^{k/2} and then square it. If k is odd, compute (A^{-1})k.

### Diagonalization

The diagonalizable matrix is expressed as

$$ A \;=\; PDP^{-1} $$

Here, **D is a diagonal matrix** and P shows the eigenvectors. Power matrix A can be expressed as:

$$ A^k \;=\; PD^k \; P^{-1} $$

You can choose any method that quickly gives you a result of the matrix power problem. You can also use our matrix to a power calculator to solve matrix exponentiation problems in seconds with a detailed solution.

## Practical Example of Matrix Power:

Let’s see a practical **example of matrix power** with its solution to understand the method used for solving power matrix problems.

**Example**:

Find A^{2},

$$ A \;=\; \left[ \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right] $$

**Solution:**

$$ A^2 \;=\; A \times A $$

$$ A \times A \;=\; \left \lfloor \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right \rfloor \times \left \lfloor \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right \rfloor $$

$$ =\; \left[ \begin{matrix} 2 \times 2 + 3 \times 0 + 1 \times 1 & 2 \times 3 + 3 \times 5 + 1 \times 1 & 2 \times 1 + 3 \times 6 + 1 \times 2 \\ 0 \times 2 + 5 \times 0 + 6 \times 1 & 0 \times 3 + 5 \times 5 + 6 \times 1 & 0 \times 1 + 5 \times 6 + 6 \times 2 \\ 1 \times 2 + 1 \times 0 + 2 \times 1 & 1 \times 3 + 1 \times 5 + 2 \times 1 & 1 \times 1 + 1 \times 6 + 2 \times 2 \\ \end{matrix} \right] $$

$$ =\; \left[ \begin{matrix} 4 + 0 + 1 & 6 + 15 + 1 & 2 + 18 + 2 \\ 0 + 0 + 6 & 0 + 25 + 6 & 0 + 30 + 12 \\ 2 + 0 + 2 & 3 + 5 + 2 & 1 + 6 + 4 \\ \end{matrix} \right] $$

$$ =\; \left[ \begin{matrix} 5 & 22 & 22 \\ 6 & 31 & 42 \\ 4 & 10 & 11 \\ \end{matrix} \right] $$

$$ A^2 \;=\; \left[ \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 5 & 22 & 22 \\ 6 & 31 & 42 \\ 4 & 10 & 11 \\ \end{matrix} \right] $$

## How to Use Matrix to a Power Calculator?

The power of a matrix calculator has a user-friendly design that enables you to easily find the value of k from the given matrix questions. Before adding the input value to the calculator, follow our instructions. These instructions are:

**Select the number**of row and column order from the given list- Enter the element of the given matrix per your selected row order number.
- Enter the value of k for the power of the matrix.
- Recheck your input matrix and k value before hitting the calculate button to start the calculation process in the matrix to power calculator.
- Click on the “Calculate” button to get the desired result for your given matrix power problems.
- If you want to try out our matrix power calculator for the first time, you can use the load example to see if it works.
- Click on the “Recalculate” button to get a new page for solving more matrix power problems and their solutions.

## Output of Power Matrix Calculator?

A matrix to power calculator gives you the **solution** to a given matrix power problem when you input it. That may include as:

**Result Option**

You can click on the result option, and it provides you with a solution to the matrix power question as per your given matrix value.

**Possible Step**

When you click on the possible steps option, it provides you with the solution of power matrix problems in steps.

## Benefits of Power of a Matrix Calculator:

The matrix to a power calculator offers many benefits when calculating matrix power questions and solutions. Benefits are:

- It is a
**manageable tool**that can be used to solve the power of a matrix in n by n order. - On the Internet, you can access a free tool that will help you solve your given matrix problems and find their power of nth order.
- Our tool saves the time and effort you spend calculating the given power problem, which takes only a few seconds.
- It is an academic tool that helps you use it for practice, gaining an in-depth knowledge of the matrix power concept.
- power matrix calculator is a trustworthy tool that provides accurate solutions according to your input value of k when you use it.
- Our matrix power calculator provides an accurate matrix power solution, which shows the efficiency of our matrix power tool whenever you use it for evaluation.