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 Ak 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 Ak/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 A2,
$$ A \;=\; \left[ \begin{matrix} 2 & 3 & 1 \\ 0 & 5 & 6 \\ 1 & 1 & 2 \\ \end{matrix} \right] $$
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.
