Introduction to Matrix Calculator:
Matrix Calculator is an online tool that helps you to find the basic arithmetic operations that are performed on the given matrix. It is used to evaluate the different order of matrices like 2 by 2 matrices, 3 by 3 matrices, or any order only if they both have the same order.
The matrix solver is an essential tool for anyone whether he is a beginner or professional who wants to learn matrices in linear algebra. It escapes you from tedious or complex calculations of matrices and provides you with quick results for your problems.
What is a Matrix?
A matrix is an array of numbers, symbols, or expressions, that are arranged in rows and columns in a rectangle shape.
Each element of a matrix denotes its row and column number. Matrices are a fundamental process in mathematics because it is used in a wide range of applications across science, engineering, and other fields.
Representation of Matrix:
The Matrix Calculator presents the matrix in its mathematical form. Mathematically it can be written as:
$$ A \;=\; \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{matrix} \right] $$
Here,
- A is the name of the matrix
- a11 to a1n is the row elements of a matrix
- a11 to am1 is the column element of a matrix
How to Calculate the Matrix?
For the calculation of matrices, the matrices calculator performs various operations like addition, subtraction, multiplication, determinant, and inverse. Let's see how to perform these basic matrix operations one by one with the help of examples.
Matrix Addition:
To add the two matrices, first, the matrix equation solver checks whether both the matrices have the same order or not. If it has the same order then it adds the corresponding elements from the matrix. Suppose that matrix addition,
$$ A \;=\; \biggr( \begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \biggr), B \;=\; \biggr( \begin{matrix} 5 & 6 \\ 7 & 8 \\ \end{matrix} \biggr) $$
Solution:
$$ A + B \;=\; \biggr( \begin{matrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 + 8 \\ \end{matrix} \biggr) \;=\; \biggr( \begin{matrix} 6 & 8 \\ 10 & 12 \\ \end{matrix} \biggr) $$
Matrix Subtraction:
For the subtraction of the matrix same addition rule is followed in which crosspounding elements of matrices are subtracted, but it should maintain the order of the matrix.
$$ A \;=\; \biggr(\begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \biggr), B \;=\; \biggr(\begin{matrix} 5 & 6 \\ 7 & 8 \\ \end{matrix} \biggr) $$
$$ A - B \;=\; \biggr(\begin{matrix} 1 - 5 & 2 - 6 \\ 3 - 7 & 4 - 8 \\ \end{matrix} \biggr) \;=\; \biggr(\begin{matrix} -4 & -4 \\ -4 & -4 \\ \end{matrix} \biggr) $$
Matrix Multiplication:
For the multiplication of two matrices A and B, the matrix calculator multiplies the elements of a row from matrix A by a column of matrix B. In the same way, the second-row elements of matrix A are multiplied by the column of matrix B. For this, suppose a matrix A and B.
$$ A \;=\; \biggr(\begin{matrix} 1 & 2 \\ 3 & 4 \\ \end{matrix} \biggr), B \;=\; \biggr(\begin{matrix} 2 & 0 \\ 1 & 2 \\ \end{matrix} \biggr) $$
$$ AB \;=\; \biggr(\begin{matrix}(1 . 2 + 2 . 1) & (1 . 0 + 2 . 2) \\ (3 . 2 + 4 . 1) & (3 . 0 + 4 . 2) \end{matrix} \biggr) $$
$$ =\; \biggr(\begin{matrix} 4 & 4 \\ 10 & 8 \\ \end{matrix} \biggr) $$
Matrix Determinant:
For the determinant of the 2 by 2 matrix the matrix solution calculator uses cross multiplication method to solve the given matrix determinant. Let's take a matrix A for finding the determinant.
$$ A \;=\; \biggr(\begin{matrix} a & b \\ c & d \\ \end{matrix} \biggr) $$
$$ det\; (A) \;=\; ad - bc $$
In the same way, if you have a 3 by 3 matrix for the determinant then you use the expansion cofactor rule to find its determinant.
Inverse of a Matrix
To find the solution of the inverse matrix, only two conditions are compulsory and the condition is when the given matrix is a square matrix and its determinant is not equal to zero. Such as:
$$ If\; det(A) \;=\; ad - cb\; ≠\; 0 $$
$$ A \;=\; \left( \begin{matrix} a & b \\ c & d \\ \end{matrix} \right) $$
$$ A^{-1} \;=\; \frac{1}{det\; (A)} \biggr(\begin{matrix} d & -b \\ -c & a \\ \end{matrix} \biggr) $$
In this method first element of the column and the second column's second element interchange their positions and the first, the second element, and the first column's first element signs have changed from positive to negative.
Transpose of a Matrix:
In the transposition of a matrix, the matrix equation calculator changes the number of rows into columns, and columns become rows in the given matrix.
$$ A \;=\; \left( \begin{matrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \\ \end{matrix} \right) $$
$$ A^{T} \;=\; \left( \begin{matrix}1 & 3 & 5 \\ 2 & 4 & 6 \\ \end{matrix} \right) $$
How to Use Matrix Calculator?
The matrix solver has a simple design so that everyone can use it to calculate the matrix problems for basic operation. You just need to put your problem in the calculator and follow some steps. These steps are:
- Choose the order of the matrix for your particular problem
- Select the method that you want to evaluate from your matrix
- Enter the elements of matrix A and B in the input field of the matrix solution calculator.
- Check the input values of A and B that you provided before clicking the calculate button to get the solution to your matrix problem.
- Click the “Calculate” button for the solution of the matrix question for a particular operation.
- If you want to check the calculation process of our matrices calculator, use the load example question to learn more about this tool.
- The Recalculate button of the matrix equation solver gives you a refreshed page for the evaluation and more examples of matrix questions to find its solution.
Final Result of Matrix Solver:
Matrix Calculator provides you a solution as per your input matrix questions when you click on the calculate button. It may contain as:
- When you click on the result button, you get the solution of the matrix for various types of operations as per your given operation on the matrix.
- The steps option provides you with the solution of the matrix questions in detail.
Benefits of the Matrices Calculator
The matrix solver has millions of benefits that you obtain whenever you use it to solve matrix problems for basic mathematical operations and get its solution immediately. These benefits are:
- The matrix equation solver is a swift tool that determines the matrix question and provides solutions in a couple of minutes without external effort.
- It is an educational tool that helps children to learn the concept of basic operations in matrix easily in just one click by sitting at home.
- The matrix algebra calculator is a handy tool that can solve matrix problems (for addition, subtraction, multiplication, rank, or inverse)
- It is a free tool that allows you to use it for the complex calculation of a matrix without any fee for its solution.
- The matrix equation calculator has an easy-to-use interface you do not need to become an expert to use it.
- Our Matrix Calculator only takes the input value and provides a solution asking too many questions about your name, email, etc.
