# Matrix Determinant Calculator

Created by Luis Fernando
Last updated: Jul 04, 2022

If you want to calculate matrix determinants, you're in the right place. This determinant solver calculates the determinant of 4x4, 3x3, and 2x2 matrices.

But what is the importance of determinants? Determinants have many applications, which we'll mention in the following section. For example, solving a 3x3 system of equations is the same as calculating the determinant of a 3x3 matrix. Keep reading to learn more about it!

## Why do we need to calculate matrix determinants?

These are some of the applications of determinants:

• For instance, we can describe systems of linear equations using matrices. The use of Cramer's rule is an example in which we use determinants to solve systems of linear equations.
• When using matrices to describe a linear transformation, it's often better to diagonalize them. We do that by calculating matrix determinants, of course.
• The determinant tells us whether the matrix has an inverse and whether we can approximate that inverse with the Moore-Penrose pseudoinverse.
• We usually need the eigenvalues of the previously mentioned transformation. To obtain them, we also need to calculate matrix determinants.

And why do we need matrices? Well, matrices describe many physical quantities, such as stress, turbulence, or the Mohr's circle.

Well, determinants are important, that's clear. Now, let's see how to calculate them.

## Calculating the determinant of 4x4, 3x3, and 2x2 matrices

The following are the formulas to calculate matrix determinants.

### Determinant of a 2x2 matrix

If

$\scriptsize A = \begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}$

then the determinant of $A$ is

$\footnotesize |A| = a_1b_2 - a_2b_1$

### Determinant of a 3x3 matrix

If

$\scriptsize B = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$

then, to calculate the determinant of such a 3x3 matrix:

$\scriptsize |B| = a_1b_2c_3 + a_2b_3c_1 + a_3b_1c_2 \\\ \ \ \ \ \ \ \ \ - a_3b_2c_1 - a_1b_3c_2 - a_2b_1c_3$

### Determinant of a 4x4 matrix

Finally:

$\scriptsize C = \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \end{bmatrix}$

then, to calculate the determinant of such a 4x4 matrix:

$\scriptsize |C| = a_1b_2c_3d_4 - a_2b_1c_3d_4 + a_3b_1c_2d_4 - \\\ a_1b_3c_2d_4 + a_2b_3c_1d_4 - a_3b_2c_1d_4 + a_3b_2c_4d_1 - \\\ a_2b_3c_4d_1 + a_4b_3c_2d_1 - a_3b_4c_2d_1 + a_2b_4c_3d_1 - \\\ a_4b_2c_3d_1 + a_4b_1c_3d_2 - a_1b_4c_3d_2 + a_3b_4c_1d_2 - \\\ a_4b_3c_1d_2 + a_1b_3c_4d_2 - a_3b_1c_4d_2 + a_2b_1c_4d_3 - \\\ a_1b_2c_4d_3 + a_4b_2c_1d_3 - a_2b_4c_1d_3 + a_1b_4c_2d_3 - \\\ a_4b_1c_2d_3$

That was a long formula.

As you can see, finding the determinant of a 3x3 and a 2x2 matrix is relatively easy, but **calculating the determinant of a 4x4 matrix is an uphill task. The best option is, undoubtedly, using our determinant solver.

Luis Fernando
Matrix size
2x2
A=
 ⌈ a₁ b₁ ⌉ ⌊ a₂ b₂ ⌋
First row
a₁
b₁
Second row
a₂
b₂
Result
Determinant |A|
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