Math Calculators, Lessons and Formulas

It is time to solve your math problem

mathportal.org

Determinant Calculator

google play badge app store badge

This calculator computes the determinant of a square matrix. The calculator shows all steps with a detailed explanation.

Tutorial

The determinant is a number that we can calculate from any square matrix.

Why do we use determinants?

We use determinants to:

1. Find out if the matrix has an inverse.

2. Find an inverse of a square matrix.

3. Solve the system of linear equations.

4. Find the characteristic polynomial, eigenvalues and eigenvectors.

Determinant 2 x 2

For any matrix with 2 rows and 2 columns, the determinant can be calculated using the formula:

determinant of a 2x2 matrix

Example

Find the determinant of $ A = \begin{bmatrix} 4 & -3 \\ 2 & 3 \end{bmatrix}$

Using the above formula, we have:

|A| = 4·3-(-3)·2 =
= 12 + 6 = 18

Determinant 3 x 3

The determinant of a 3x3 matrix can be solved using the determinants of the 2x2 matrices using the formula:

formula for calculating determinant of a 3x3 matrix

Example

Find the determinant of $ A = \begin{bmatrix} 2 & 0 & 6 \\ 3 & 2 & 1 \\ 0 & 4 & 8 \end{bmatrix}$

Using the determinant formula for 3x3 matrices, we have

$$\begin{aligned} A &= \begin{bmatrix} 2 & 0 & 6 \\ 3 & 2 & 1 \\ 0 & 4 & 8 \end{bmatrix} = \\[2 em] &= 2 \cdot \begin{bmatrix} 2 & 1 \\ 4 & 8 \end{bmatrix} - 0 \cdot \begin{bmatrix} 3 & 2 \\ 1 & 8 \end{bmatrix} + 6 \cdot \begin{bmatrix} 3 & 2 \\ 0 & 4 \end{bmatrix} = \\[2 em] &= 2 \cdot 12 - 0 \cdot 22 + 6 \cdot 12 = 96 \end{aligned} $$
Determinant calculator
with step-by-step explanations
help ↓↓ examples ↓↓
2 x 2
3 x 3
4 x 4
5 x 5
thumb_up 186 thumb_down

Get Widget Code

Examples
ex 1:
What is the determinant of the matrix $$ A = \left[ \begin{array}{cc} -\frac{3}{2} & 2 \\ -5 & 4 \end{array} \right]$$
ex 2:
Find the determinant of the matrix $$ A = \left[ \begin{array}{cc} 4 & 8 & 5 \\ -3 & 2 & 6 \\ 11 & -3 & 5 \end{array} \right]$$
ex 3:
Compute determinant $$ A = \left[ \begin{array}{cc} -1 & 2 & 4 & 1 \\ 5 & 3 & 1 & 1 \\ 3 & 7 & 9 & 3 \\ 2 & -1 & 2 & 4 \end{array} \right]$$
Find more worked-out examples in our database of solved problems.
452 861 664 solved problems
working...
×
ans:
syntax error
C
DEL
ANS
±
(
)
÷
×
7
8
9
4
5
6
+
1
2
3
=
0
.