Determinant of a Matrix Calculator

Now you can calculate the determinant of the square matrix easily because of the determinant of a matrix calculator with steps.

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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.

Determinant of a matrix Calculator with Steps

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.

Related References

What is matrix determinant? Explain
Give some examples of determinant of a matrix:
How to evaluate determinant of matrix?
An overview of matrix determinant:

Frequently Ask Questions

What happens if the determinant of a matrix is 0?

When the determinant of a matrix is 0, the matrix is singular, indicating linear dependence among rows or columns, and has profound implications in linear algebra, that provide uniqueness in solution, and geometric interpretation.

If the determinant of a matrix is 0, several important properties and implications arise. A matrix whose determinant gives 0, that matrix is called a singular matrix. Singular matrices do not have an inverse.

For the determinant of any matrix, the rows and columns of the matrix should be linearly dependent. This means that at least one row or column can be expressed as a linear combination of the others. It may be the eigenvalue of the matrix is 0.

What happens to the determinant when the matrix is transposed?

The determinant of a matrix is invariant under transposition, which means the determinant of a matrix is equal to the determinant of its transpose. This property simplifies the calculation process in linear algebra.

The determinant of a matrix does not change its position when you take the transposition of that matrix. This is an important property of determinants and can be expressed as:

$$ det(AT) \;=\; \frac{det(A)}{(A^T)} $$

Where A is any square matrix, and A^T is the transpose of A.

What is the difference between a determinant and a matrix?

Determinants and matrices are important concepts in linear algebra, that serve different objects and have distinct properties. Their difference is:

Matrix:

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices are often used to represent linear transformations, systems of linear equations, and data sets.

It has different types such as square matrix, rectangular matrix, identity matrix, zero matrix

Determinant:

The determinant is a scalar value that can be evaluated from the elements of a square matrix. It has similar properties to the matrix. The determinant of a matrix A is denoted as det(A) or ∣A∣.

If det(A) = 0 the matrix A is singular (non-invertible). If det(A)≠0 the matrix A is non-singular (invertible).

The determinant of a product of matrices is the product of their determinants: $$ det(AB) \;=\; det(A) ⋅ det(B) $$. Hence, a matrix is a is method that used in a wide range of operations and applications, while a determinant is a specific value that provides insights into the properties of square matrices.

What is the determinant of an inverse matrix?

The determinant of an inverse matrix A is the reciprocal of the determinant of A. This relationship provides a simple way to determine the effect of a matrix on its determinant, which ensures consistency in linear algebra operations.

The determinant and inverse of a matrix have a specific relationship, If A is a square matrix and A⁻¹ is its inverse, the relationship can be expressed as:

$$ det(A . A^{-1}) \;=\; det(I) \;=\; I $$

$$ det(A) . det(A^{-1}) \;=\; 1 $$

$$ det(A^{-1}) \;=\; \frac{1}{det(A)} $$

What happens to the determinant when the matrix is squared?

When the matrix is squared matrix A², the determinant of that square matrix becomes the original matrix A. It means square matrix A is the multiply of itself as to obtain A².

The determinant of the squared matrix A² has a specific relationship that simplifies the calculation of determinants for powers of matrices and helps in understanding the effect of matrix operations on determinants. Its relationship is described as:

$$ det(A^2) \;=\; (det(A))^2 $$

Amanda Jepson

Last Updated February 21, 2024