## Introduction to the Determinant of a Matrix Calculator:

Determinant of a matrix calculator is an online tool that is designed to **compute the determinant** of a given square matrix. In a few seconds, you can use this tool to evaluate the determinant of 2 by 2 and 3 by 3 matrix order.

The determinant of a 4x4 matrix calculator is a valuable educational and practical tool in the field of mathematics and beyond. It streamlines the process of finding determinants, aiding in the understanding and application of linear algebra concepts.

## What is the Determinant of a Matrix?

Matrix determinant is a fundamental concept in linear algebra which is used to find a **square matrix's elements**. It is used in solving systems of linear equations, finding the inverse of a matrix, and determining the volume scaling factor of the linear transformation in the matrix.

Determinant methods are widely used in linear algebra, differential equations, and multivariable calculus.

## Rules Used for Determinant Matrix:

Determinant matrix has different methods for finding the determinant of 2 by 2 order of matrix and 3 by 3 matrix. The result of the determinant matrix has always a scalar value.

For 2 by 2 matrix: the determinant of a matrix calculator used simply a **cross multiplication method** to find the determinant of A,

$$ A \;=\; \biggr( \begin{matrix} a & b \\ c & d \\ \end{matrix} \biggr) $$

$$ det(A) \;=\; ad - bc $$

For 3 by 3 matrix: the matrix determinant calculator with steps used the **expansion of cofactor** method for 3 by 3 matrix

$$ B \;=\; \biggr(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} \biggr) $$

$$ det(B) \;=\; a(ei - fh) - b(di - fg) + c(dh - eg) $$

## How to Find the Determinant of a Matrix?

The determinant of a matrix provides a value that can be **calculated from the elements** of a square matrix A. The determinant provides critical insights into the matrix's properties and is a fundamental concept in linear algebra with various applications.

The method used by the matrix determinant calculator with steps to find the determinant varies depending on the size of the matrix. For this let us observe two examples of a 2 by 2 matrix and a 3 by 3 matrix where two different methods are used to find the determinant.

## How to Find the Determinant of a 3x3 Matrix?

To find the determinant of the 3 by 3 matrix, let's take a practical example to understand how the determinant of a matrix calculator works.

Find the determinant of A,

$$ A \;=\; \biggr[\begin{matrix} 3 & 2 & 1 \\ 0 & 1 & -2 \\ 1 & 3 & 4 \\ \end{matrix} \biggr] $$

**Solution**:

Write the given matrix into determinant form,

$$ |A| \;=\; \biggr|\begin{matrix} 3 & 2 & 1 \\ 0 & 1 & -2 \\ 1 & 3 & 4 \\ \end{matrix} \biggr| $$

Expand the first row or the first column along with its cofactors,

$$ =\; 3 \biggr|\begin{matrix} 1 & -2 \\ 3 & 4 \\ \end{matrix} \biggr| -2 \biggr| 0 & -2 \\ 1 & 4 \\ \end{matrix} + 1 \biggr|\begin{matrix} 0 & 1 \\ 1 & 3 \\ \end{matrix} \biggr| $$

Simplify the given expression to the cofactor of the determinant,

$$ =\; 3(4 + 6) -2(0 + 2) + 1(0 - 1) $$

$$ =\; 30 - 4 - 1 $$

The solution of the determinant matrix in 3 by 3 order is 25.

$$ |A| \;=\; 25 $$

## How to Find the Determinant of a 2x2 Matrix?

To find the determinant of the 2 by 2 matrix, the determinant of matrix calculator uses a simple cross-multiplication method such as:

Find the determinant of A,

$$ A \;=\; \biggr[\begin{matrix} 3 & 1 \\ -2 & 3 \\ \end{matrix} \biggr] $$

**Solution**:

Convert the given matrix into determinant form

$$ |A| \;=\; \biggr|\begin{matrix} 3 & 1 \\ -2 & 3 \\ \end{matrix} \biggr| $$

Then cross multiply the given element inside the determinant.

Simply the given expression to get the solution of the given square matrix.

$$ =\; 9 -(-2) \;=\; 9 + 2 \;=\; 11 $$

Therefore the solution of the given 2 by 2 matrix solution is 11.

## How to Use the Determinant of a Matrix Calculator?

The matrix determinant calculator with steps has a simple design so, everyone can use it to calculate the determinant of the given matrix. Before adding the input value, you must follow some instructions. These instructions are:

- Select the size of the matrix as per the given matrix order.
- Enter the elements of the given matrix in the input box of the determinant matrix calculator to find the value of the determinant.
- Review your input value before hitting the calculate button to start the calculation process for finding the solution of the matrix determinant.
- Click on the “Calculate” button to get the required result of the matrix determinant problem.
- If you are trying out our determinant of 3x3 matrix calculator for the first time then you can use the load example
- Click on the “Recalculate” button to get a new page for solving more Matrix Determinant questions.

## Final Result of Determinant of a 4x4 Matrix Calculator:

The determinant of a matrix calculator gives you the solution to a given matrix problem when you give the input value to it. It may be included as:

**Result Option**

You can click on the result option and it provides you with a solution for the matrix determinant questions

**Possible Step**

Possible steps option of the determinant of 3x3 matrix calculator provides you with the solution in which evaluation steps for matrix determinant questions.

## Useful Features of Determinant of Matrix Calculator:

The determinant matrix calculator has many useful features that help you to get the solution of the matrix determinant problems whenever you use it. These features are:

- The determinant of a 3x3 matrix calculator saves your time and effort from doing lengthy and complex calculations of the determinant matrix question in less than a minute
- It is a free tool so that you can use it to find the determinant question without paying any fee.
- The determinant of a 4x4 matrix calculator is an easy-to-use tool so you do not need any type of technical expertise just use it to easily calculate the given matrix problems.
- It is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate the matrix determinant problem.
- The determinant of a matrix calculator provides you with a solution procedure in a step-by-step method for more clarity.