# Determinant Calculations

## Calculator

Determinant Calculators with step by step calculation of the determinant value.

Determinant 2x2

Step by step solution with Sarrus Rule, Laplace Expansion and Gaussian Method.

Determinant 3x3 Determinant 4x4 Determinant 5x5 Determinant NxN

Symbolic Determinant Calculators with step by step calculation of the determinant value.

Determinant 3x3 Determinant 4x4 Determinant 5x5

### Determinant History

Determinants historically considered before the matrices. Originally a determinant was defined as a property of a system of linear equations. The determinant "determines" whether the system of equations has a unique solution (this is exactly the case if the determinant is non-zero). In this context, 2x2 matrices were treated by Cardano at the end of the 16th century and larger matrices by Leibniz about 100 years later.

### Determinant

Each square matrix can be assigned a unique number, which is called the determinant (det(A)) of the matrix. In general, the determinant of an NxN matrix is defined by the Leibniz formula:

$det A= ∑ σ ∈ Sn sgn σ Π i = 1 n Aiρi$

here the sum has to be extended over all the permutations σ. Thus, from the elements of A, all possible products are formed for each n-element in such a way that each of the products of each row and column contains exactly one element. These products are added and the sum is the determinant of A. The sign of the summands is positive for even permutations and negative for odd permutations.

## Determinant Calculation Rules

### Interchanging two Rows of the determinant

The interchanging two rows of the determinant changes only the sign and not the value of the determinant.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{k1}& {a}_{k2}& \dots & {a}_{kn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|={-}\phantom{\rule{0.3em}{0ex}}|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{k1}& {a}_{k2}& \dots & {a}_{kn}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|$

### Interchanging two Columns of the determinant

The interchanging two columns of the determinant changes only the sign and not the value of the determinant.

$\mathrm{det A}=|\begin{array}{ccccccc}{a}_{11}& \dots & {a}_{1j}& \dots & {a}_{1k}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {a}_{2j}& \dots & {a}_{2k}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {a}_{nj}& \dots & {a}_{nk}& \dots & {a}_{nn}\end{array}|={-}\phantom{\rule{0.3em}{0ex}}|\begin{array}{ccccccc}{a}_{11}& \dots & {a}_{1k}& \dots & {a}_{1j}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {a}_{2k}& \dots & {a}_{2j}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {a}_{nk}& \dots & {a}_{nj}& \dots & {a}_{nn}\end{array}|$

### Factor in a Row of the determinant

Extracting a common factor from a row. A common factor in all elements of a row can be drawn as a multiplier before the determinate. The value of the determinate is then obtained from the multiplication of the factor with the value of the resulting determinate det A'.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{j1}& {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{j2}& \dots & {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}\mathrm{det A\text{'}}$

Extracting a common factor from a column. A common factor in all elements of a column can be drawn as a multiplier before the determinate. The value of the determinate is then obtained from the multiplication of the factor with the value of the resulting determinate det A'.

$\mathrm{det A}=|\begin{array}{ccccc}{a}_{11}& \dots & {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{1j}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{2j}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{nj}& \dots & {a}_{nn}\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}|\begin{array}{ccccc}{a}_{11}& \dots & {a}_{1j}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {a}_{2j}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {a}_{nj}& \dots & {a}_{nn}\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}\mathrm{det A\text{'}}$

### Addition of Rows

Addition of a row of the determinant with the multiple of another row. The value of a determinant does not change when a multiple of another row is added to the row.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{k1}& {a}_{k2}& \dots & {a}_{kn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{k1}& {a}_{j2}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{k2}& \dots & {a}_{jn}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{kn}\\ & ⋮\\ {a}_{k1}& {a}_{k2}& \dots & {a}_{kn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|$

### Addition of Columns

Addition of a column of the determinant with the multiple of another column. The value of a determinant does not change when a multiple of another column is added to the column.

$\mathrm{det A}=|\begin{array}{ccccccc}{a}_{11}& \dots & {a}_{1j}& \dots & {a}_{1k}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {a}_{2j}& \dots & {a}_{2k}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {a}_{nj}& \dots & {a}_{nk}& \dots & {a}_{nn}\end{array}|=|\begin{array}{ccccccc}{a}_{11}& \dots & {a}_{1j}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{1k}& \dots & {a}_{1k}& \dots & {a}_{1n}\\ {a}_{21}& \dots & {a}_{2j}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{2k}& \dots & {a}_{2k}& \dots & {a}_{2n}\\ & ⋮\\ {a}_{n1}& \dots & {a}_{nj}+{\lambda }\phantom{\rule{0.3em}{0ex}}{a}_{nk}& \dots & {a}_{nk}& \dots & {a}_{nn}\end{array}|$

### Multiplication Theorem

The determinant of the product of two matrices is the product of the determinants of the matrices.

$det\left(A\cdot B\right)=det\left(A\right)\cdot det\left(B\right)$

It also follows the following relationship.

$det\left({A}^{k}\right)={det\left(A\right)}^{k}$

### Transposition Theorem

The determinant of a transpose matrix is equal to the determinant of the matrix itself.

$det\left({A}^{T}\right)=det\left(A\right)$

### Inverse Matrix

The determinant of the inverse of a matrix is ​​equal to the reciprocal of the determinant of the matrix itself

$det\left({A}^{\mathrm{-1}}\right)={det\left(A\right)}^{\mathrm{-1}}=\frac{1}{\mathrm{det A}}$

### Box Theorem

Has a determinant following box structur with square boxes B and D, then lets its determinant as the product of the determinants of B and D directions.

$\mathrm{det A}=|\begin{array}{cc}B& C\\ 0& D\end{array}|=det\left(B\right)det\left(D\right)\mathrm{det A}=|\begin{array}{cc}B& 0\\ C& D\end{array}|=det\left(B\right)det\left(D\right)$

## Calculating the Value of the Determinant

### Determinant of 0x0 Matrix

The determinant of a 0x0 matrix is defined as 1.

### Determinant of 1x1 Matrix

A 1x1-matrix is a matrix with only one element and the determinant is given by the element itself.

$\mathrm{det A}=|\begin{array}{c}{a}_{11}\end{array}|={a}_{11}$

### Determinant of 2x2 Matrix

For a 2x2-matrix, the determinant is calculated as follows.

$\mathrm{det A}=|\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}|={a}_{11}{a}_{22}-{a}_{21}{a}_{12}$

### Determinant of 3x3 Matrix

For the calculation of 3x3 determinant there are different ways. With the Lapace development it is possibel to reduce the determinant to 2x2 determinantes. A direct way to compute the determinant is the Sarrus Rule. The Sarrus rule states that the determinant of a square 3x3 matrix is calculated by subtracting the sum of the products of the main diagonals from the sum of the products of the secondary diagonals.

### Determinant of a 3x3 matrix according to the Sarrus Rule

The determinant is calculated as follows by the Sarrus Rule. Schematically, the first two columns of the determinant are repeated so that the major and minor diagonals can be virtual connected by a linear line. Then one makes the products of the main diagonal elements and adds this products. With the secondary diagonals you shall do the same. The difference between the two gives the determinant of the matrix.

$\mathrm{det A}=|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|\phantom{\rule{0.5em}{0ex}}\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\\ {a}_{31}& {a}_{32}\end{array}{|}\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\\ ={a}_{11}{a}_{22}{a}_{33}+{a}_{12}{a}_{23}{a}_{31}+{a}_{33}{a}_{21}{a}_{32}\\ \phantom{\rule{0.3em}{0ex}}\end{array}\begin{array}{c}\phantom{\rule{0.3em}{0ex}}\\ -\left({a}_{31}{a}_{22}{a}_{13}+{a}_{32}{a}_{23}{a}_{11}+{a}_{33}{a}_{21}{a}_{12}\right)\\ \phantom{\rule{0.3em}{0ex}}\end{array}$

## Determinant of NxN Matrix

### Laplace Expansion Theorem

The Laplacian development theorem provides a method for calculating the determinant, in which the determinant is developed after a row or column. The dimension is reduced and can be reduced further step by step up to a scalar.

$\mathrm{det A}=\sum _{i=1}^{n}{-1}^{i+j}\cdot {a}_{ij}\mathrm{det}{A}_{ij}\text{( Expansion on the j-th column )}$

$\mathrm{det A}=\sum _{j=1}^{n}{-1}^{i+j}\cdot {a}_{ij}\mathrm{det}{A}_{ij}\text{( Expansion on the i-th row )}$

where Aij, the sub-matrix of A, which arises when the i-th row and the j-th column are removed.

### Example of the Laplace expansion according to the first row on a 3x3 Matrix.

$\mathrm{det A}=|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|$

The first element is given by the factor a11 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{11}|\begin{array}{cc}{a}_{22}& {a}_{23}\\ {a}_{32}& {a}_{33}\end{array}|$

The second element is given by the factor a12 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{12}|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}|$

The third element is given by the factor a13 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{13}|\begin{array}{cc}{a}_{21}& {a}_{22}\\ {a}_{31}& {a}_{32}\end{array}|$

With the three elements the determinant can be written as a sum of 2x2 determinants.

$\mathrm{det A}=|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|={a}_{11}|\begin{array}{cc}{a}_{22}& {a}_{23}\\ {a}_{32}& {a}_{33}\end{array}|-{a}_{12}|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}|+{a}_{13}|\begin{array}{cc}{a}_{21}& {a}_{22}\\ {a}_{31}& {a}_{32}\end{array}|$

It is important to consider that the sign of the elements alternate in the following manner.

$|\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}|$

### Example of the Laplace expansion according to the second column on a 3x3 Matrix.

$\mathrm{det A}=|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|$

The first element is given by the factor a12 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{12}|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}|$

The second element is given by the factor a22 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{22}|\begin{array}{cc}{a}_{11}& {a}_{13}\\ {a}_{31}& {a}_{33}\end{array}|$

The third element is given by the factor a23 and the sub-determinant consisting of the elements with green background.

$|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=>{a}_{23}|\begin{array}{cc}{a}_{11}& {a}_{13}\\ {a}_{21}& {a}_{23}\end{array}|$

With the three elements the determinant can be written as a sum of 2x2 determinants.

$\mathrm{det A}=|\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}|=-{a}_{12}|\begin{array}{cc}{a}_{21}& {a}_{23}\\ {a}_{31}& {a}_{33}\end{array}|+{a}_{22}|\begin{array}{cc}{a}_{11}& {a}_{13}\\ {a}_{31}& {a}_{33}\end{array}|-{a}_{32}|\begin{array}{cc}{a}_{11}& {a}_{13}\\ {a}_{21}& {a}_{23}\end{array}|$

It is important to consider that the sign of the elements alternate in the following manner.

$|\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}|$

### Example of the expansion according to the j-th row of a NxN determinant.

The Laplace expansion reduces the NxN determinant to a sum of (N-1)x(N-1) determinants.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{j1}& {a}_{j2}& \dots & {a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|=±{a}_{j1}|\begin{array}{ccc}{a}_{12}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{\mathrm{j-1}2}& \dots & {a}_{\mathrm{j-1}n}\\ {a}_{\mathrm{j+1}2}& \dots & {a}_{\mathrm{j+1}n}\\ & ⋮\\ {a}_{n2}& \dots & {a}_{nn}\end{array}|±{a}_{j2}|\begin{array}{cccc}{a}_{11}& {a}_{13}& \dots & {a}_{1n}\\ & ⋮\\ {a}_{\mathrm{j-1}1}& {a}_{\mathrm{j-1}3}& \dots & {a}_{\mathrm{j-1}n}\\ {a}_{\mathrm{j+1}1}& {a}_{\mathrm{j+1}3}& \dots & {a}_{\mathrm{j+1}n}\\ & ⋮\\ {a}_{n1}& {a}_{n3}& \dots & {a}_{nn}\end{array}|±\dots ±{a}_{jn}|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1\mathrm{n-1}}\\ & ⋮\\ {a}_{\mathrm{j-1}1}& {a}_{\mathrm{j-1}2}& \dots & {a}_{\mathrm{j-1}\mathrm{n-1}}\\ {a}_{\mathrm{j+1}1}& {a}_{\mathrm{j+1}2}& \dots & {a}_{\mathrm{j+1}\mathrm{n-1}}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{n\mathrm{n-1}}\end{array}|$

### Gauss Method

With the Gauss method, the determinant is so transformed that the elements of the lower triangle matrix become zero. To do this, you use the row-factor rules and the addition of rows. The addition of rows does not change the value of the determinate. Factors of a row must be considered as multipliers before the determinat. If the determinat is triangular and the main diagonal elements are equal to one, the factor before the determinant corresponds to the value of the determinant itself.

$\mathrm{det A}=|\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1n}\\ {a}_{j1}& \phantom{\rule{0.3em}{0ex}}{a}_{j2}& \dots & \phantom{\rule{0.3em}{0ex}}{a}_{jn}\\ & ⋮\\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nn}\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}|\begin{array}{cccc}1& {a}_{12}& \dots & {a}_{1n}\\ 0& 1& \dots & {a}_{jn}\\ & ⋮\\ 0& 0& \dots & 1\end{array}|={\lambda }\phantom{\rule{0.3em}{0ex}}\mathrm{det A\text{'}}={\lambda }$

### Cramers Rule

The Cramers rule uses determiants to solve a system of linear equations. For the case of a linear (N×N) system of equations with det(A) not equal to 0, the solution can be expressed in the following form:

$x={A}^{-1}b$

${x}_{i}=\frac{1}{\mathrm{det A}}|\begin{array}{c}{a}_{11}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{1}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{1n}\\ {a}_{21}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{2}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{2n}\\ ⋮\\ {a}_{n1}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{b}_{n}\phantom{\rule{1em}{0ex}}\dots \phantom{\rule{1em}{0ex}}{a}_{nn}\end{array}|$

${x}_{i}=\frac{{D}_{i}}{D}$

The determinant in the numerator Di from D = det A is shown by the i-th column in D is replaced by b.

### Definitions

The matrix A is called Regular if the determinant of A is not equal to 0.

The matrix A is called Singular if the determinant of A is equal to 0.

The matrix A is Invertible if the determinant of A is not equal to 0.

Conclusions from the multiplication theorem:

$det\left(A\cdot B\right)=det\left(B\cdot A\right)$

$det\left({C}^{\mathrm{-1}}AC\right)=det\left(A\right)$

## More Calculators

Here is a list of of further useful calculators:

Index Matrix Determinant Determinant 2x2 Determinant 3x3 Determinant 3x3 symbolic Determinant 4x4 Determinant 4x4 symbolic Determinant 5x5 Determinant NxN