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Table of Content
 1 What is a Determinant? 2 How to Find the Determinant of a Matrix Manually (Step-by-Step): 3 Determinant Properties: 4 What are determinants used for? 5 What does a determinant of 0 mean? 6 Can a determinant be negative?
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An online determinant calculator helps you to compute the determinant of the given matrix input elements. This calculator determines the matrix determinant value up to 5×5 size of matrix. It is calculated by multiplying its main diagonal members & reducing matrix to row echelon form. We have detailed information on how to calculate it manually, definition, formulas and many other useful data related to the determinant of the matrix. Our calculator determines the result with following different calculation methods:

• Expand along column.
• Expand along row.
• Leibniz formula.
• Triangle’s rule.
• Rule of Sarrus.

## What is a Determinant?

It is a scalar value that is obtained from the elements of the square matrix and having the certain properties of the linear transformation described by the matrix. The determinant of a matrix is positive or negative depend on whether linear transformation preserves or reverses the orientation of a vector space. It helps us to find the inverse of the matrix as well as the things that are useful in the systems of linear equations, calculus & more. It is denoted as det (A), det A, or |A|.

Note:

Matrices are enclosed in the square brackets while the determinants are denoted with the vertical bars. Matrix is an array of numbers but the determinant is a single number.

## How to Find the Determinant of a Matrix Manually (Step-by-Step):

The determinant of the matrices can be calculated from the different methods. Here we give the detailed formulas for different order of matrix to find the determinant from different methods:

### For 2×2 Matrix Multiplications:

No matter, which method you selected for the calculations, the determinant of matrix A = (aij)2×2 is determined by the following formula:

$$det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \\$$

$$det⁡ A = ad-bc$$

Example:
Find determinant of 2×2 matrix A

$$det A = \begin{vmatrix} 4 & 12 \\ 2 & 7 \end{vmatrix} \\$$

Solution:

$$det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \\$$

$$|A| = (7)(4) – (2)(12)$$
$$|A| = 28 – 24$$
$$|A| = 4$$

### For 3×3 Matrix Multiplications:

The calculations for 3×3 matrixes from different methods are discussed here:

#### Expand Along the Column:

For the calculations of matrix A = (aij)3×3 from expansion of column is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c\\d & e & f \\g & h & i \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} e & f \\h & i\end{vmatrix} – d\begin{vmatrix}b & c \\h & i\end{vmatrix}+g\begin{vmatrix}b & c \\e & f\end{vmatrix}$$

Example:
Find

$$det A = \begin{vmatrix} 2 & 0 & 3\\1 & 4 & 1 \\0 & 4 & 7 \end{vmatrix} \\$$?

Solution:

$$det⁡ A= 2\begin{vmatrix} 4 & 1 \\4 & 7\end{vmatrix} – 1\begin{vmatrix}0 & 3 \\4 & 7\end{vmatrix}+0\begin{vmatrix}0 & 3 \\4 & 1\end{vmatrix}$$

$$det⁡ A = 2[(7)(4)-(4)(1)]-1[(4)(3)-(7)(0)]+ 0[(4)(3)-(1)(0)]$$

$$det⁡ A = 2[28-4]-1[12-0]+ 0[12-0]$$

$$det⁡ A = 2-1+ 0$$

$$det⁡ A = 48-12+ 0$$

$$det⁡ A = 36$$

#### Expand Along the Row:

For the calculations of matrix A = (aij)3×3 from expansion of row is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c\\d & e & f \\g & h & i \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} e & f \\h & i\end{vmatrix} – b\begin{vmatrix}d & f \\g & i\end{vmatrix}+c\begin{vmatrix}d & e \\g & h\end{vmatrix}$$

Example:

Find

$$det A = \begin{vmatrix} 3 & 0 & 2\\1 & 4 & 1 \\7 & 0 & 4 \end{vmatrix} \\$$?

Solution:

$$det⁡ A= 3\begin{vmatrix} 4 & 1 \\0 & 4\end{vmatrix} – 0\begin{vmatrix}1 & 1 \\7 & 4\end{vmatrix}+2\begin{vmatrix}1 & 4 \\7 & 0\end{vmatrix}$$

$$det⁡ A = 3[(4)(4)-(0)(1)]-0[(4)(1)-(7)(1)]+ 2[(0)(1)-(7)(4)]$$
$$det⁡ A = 3[16-0]-0[4-7]+ 2[0-28]$$
$$det⁡ A = 3-0[-3]+ 2[-28]$$
$$det⁡ A = 48+0- 56$$
$$det⁡ A = -8$$

#### Leibniz Formula:

For the calculations of matrix A = (aij)3×3 by using Leibniz formula is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c\\d & e & f \\g & h & i \end{vmatrix} \\$$

$$det⁡ A =(a*e*i)-(a*f*h)-(b*d*i)+(b*f*g)+(c*d*h)-(c*e*g)$$

Example:

Find

$$det A = \begin{vmatrix} 2 & 3 & 8\\6 & 1 & 2 \\5 & 8 & 9 \end{vmatrix} \\$$?

Solution:

$$det A = \begin{vmatrix} 2 & 3 & 8\\6 & 1 & 2 \\5 & 8 & 9 \end{vmatrix} \\$$
$$det⁡ A = 2*1*9-2*2*8-3*6*9+3*2*5+8*6*8-8*1*5$$

$$det A =198$$

#### Triangle’s Rule:

For the calculations of matrix A = (aij)3×3 from Triangle’s rule is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c\\d & e & f \\g & h & i \end{vmatrix} \\$$

Image

$$det⁡ A =(a*e*i)-(a*f*h)-(b*d*i)+(b*f*g)+(c*d*h)-(c*e*g)$$

Example:

Find

$$det A = \begin{vmatrix} 4 & 5 & 8\\0 & 4 & 9 \\1 & 2 & 3 \end{vmatrix} \\$$?

Solution:

$$det A = \begin{vmatrix} 4 & 5 & 8\\0 & 4 & 9 \\1 & 2 & 3 \end{vmatrix} \\$$
$$det⁡ A = 4*4*3+5*9*1+8*0*2-1*4*8-2*9*4-3*0*5$$

$$det A =-11$$

#### Rule of Sarrus:

For the calculations of matrix A = (aij)3×3 by Rule of Sarrus is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c\\d & e & f \\g & h & i \end{vmatrix} \\$$

Image

$$det⁡ A =(a*e*i)-(a*f*h)-(b*d*i)+(b*f*g)+(c*d*h)-(c*e*g)$$

Example:

Find

$$det A = \begin{vmatrix} 9 & 5 & 1\\3 & 5 & 7 \\4 & 8 & 6 \end{vmatrix} \\$$?

Solution:

$$det A = \begin{vmatrix} 9 & 5 & 1\\3 & 5 & 7 \\4 & 8 & 6 \end{vmatrix} \\$$

$$det⁡ A = 9*5*6+5*7*4+1*3*8-4*5*1-8*7*9-6*3*5$$

$$det A = -180$$

Our determinant calculator calculates the determinant of given matrix by all the formulas given but it generate results by Leibniz & Sarrus till the 3×3 matrix.

### For 4×4 Matrix Multiplications:

The calculations for 4×4 matrixes from different methods are discussed here:

#### Expand Along the Column:

For the calculations of matrix A = (aij)4×4 from expansion of column is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c & d\\e & f & g &h \\i & j & k & l \\ m & n & o & p \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} f & g & h\\j & k & l\\n & o & p\end{vmatrix} – e\begin{vmatrix}b & c & d\\j & k & l\\ n & o & p\end{vmatrix}+i\begin{vmatrix}b & c & d \\f & g & h\\n & o & p\end{vmatrix}-m\begin{vmatrix}b & c & d\\f & g & h\\j & k & l\end {vmatrix}$$

Then, simply determine the determinant of 3×3 by using above formula of 3×3.

Example:

Find

$$det A = \begin{vmatrix} 1 & 8 & 7 & 2\\2 & 4 & 3 &8 \\1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{vmatrix} \\$$?

Solution:

$$det⁡ A= 1\begin{vmatrix}4 & 3 & 8\\4 & 3 & 2\\4 & 9 & 6\end{vmatrix} – 2\begin{vmatrix}8 & 7 & 2\\4 & 3 & 2\\ 4 & 9 & 6\end{vmatrix}+1\begin{vmatrix}8 & 7 & 2 \\4 & 3 & 8\\4 & 9 & 6\end{vmatrix}-1\begin{vmatrix}8 & 7 & 2\\4 & 3 & 8\\4 & 3 & 2\end {vmatrix}$$

$$det⁡ A=1( 4\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix} – 3\begin{vmatrix}4 & 2 \\4 & 6\end{vmatrix}+8\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) -2( 8\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix} – 7\begin{vmatrix}4 & 2 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) +1( 8\begin{vmatrix}3 & 8 \\9 & 6\end{vmatrix} – 7\begin{vmatrix}4 & 8 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) -1( 8\begin{vmatrix} 3 & 8 \\3 & 2\end{vmatrix} – 7\begin{vmatrix}4 & 8 \\4 & 6\end{vmatrix}+2\begin{vmatrix}4 & 3 \\4 & 3\end{vmatrix})$$

$$det⁡ A = 1[4(18-18)-3(24-8)+ 8(36-12)]-2[ 8(18-18)-7(24-8)+ 2(36-12)]+ 1[ 8(18-72)-7(24-32)+ 2(36-12)] -1[8(6-24)-7(8-32)+ 2(12-12)]$$

$$det⁡ A = 1[4(0)-3(16)+ 8(24)]-2[ 8(0)-7(16)+ 2(24)]+ 1[ 8(-54)-7(-8)+ 2(24)]-1[8(-18)-7(-24)+ 2(0)]$$

$$det⁡ A = 1[0-48+192]-2[0-112+48]+ 1[ -432+56+48]-1[-144+168+0]$$

$$det⁡ A = 1-2[-64]+ 1[-328]-1$$

$$det⁡ A = 144+128-328- 24$$

$$det⁡ A = -80$$

#### Expand Along the Row:

For the calculations of matrix A = (aij)4×4 from expansion of row is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c & d\\e & f & g &h \\i & j & k & l \\ m & n & o & p \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} f & g & h\\j & k & l\\n & o & p\end{vmatrix} – b\begin{vmatrix}e & g & h\\i & k & l\\ m & o & p\end{vmatrix}+c\begin{vmatrix}e & f & h \\i & j & l\\m & n & p\end{vmatrix}-d\begin{vmatrix}e & f & g\\i & j & k\\m & n & o\end {vmatrix}$$

Then, simply determine the determinant of 3×3 by using above formula of 3×3.
Example:
Find

$$det A = \begin{vmatrix} 1 & 8 & 7 & 2\\2 & 4 & 3 &8 \\1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{vmatrix} \\$$?
Solution:

$$det⁡ A= 1\begin{vmatrix}4 & 3 & 8\\4 & 3 & 2\\4 & 9 & 6\end{vmatrix} – 8\begin{vmatrix}2 & 3 & 8\\1 & 3 & 2\\ 1 & 9 & 6\end{vmatrix}+7\begin{vmatrix}2 & 4 & 8 \\1 & 4 & 2\\1 & 4 & 6\end{vmatrix}-2\begin{vmatrix}2 & 4 & 3\\1 & 4 & 3\\1 & 4 & 9\end {vmatrix}$$

$$det⁡ A=1( 4\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix} – 3\begin{vmatrix}4 & 2 \\4 & 6\end{vmatrix}+8\begin{vmatrix}4 & 3 \\4 & 9\end{vmatrix}) -8( 2\begin{vmatrix} 3 & 2 \\9 & 6\end{vmatrix} – 3\begin{vmatrix}1 & 2 \\1 & 6\end{vmatrix}+8\begin{vmatrix}1 & 3 \\1 & 9\end{vmatrix}) +7( 2\begin{vmatrix} 4 & 2 \\4 & 6\end{vmatrix} – 4\begin{vmatrix}1 & 2 \\1 & 6\end{vmatrix}+8\begin{vmatrix}1 & 4 \\1 & 4\end{vmatrix}) -2( 2\begin{vmatrix} 4 & 3 \\4 & 9\end{vmatrix} – 4\begin{vmatrix}1 & 3 \\1 & 9\end{vmatrix}+3\begin{vmatrix}1 & 4 \\1 & 4\end{vmatrix})$$

$$det⁡ A = 1[4(18-18)-3(24-8)+ 8(36-12)]-8[ 2(18-18)-3(6-2)+ 8(9-3)]+ 7[ 2(24-8)-4(6-2)+ 8(4-4)]-2[2(36-12)-4(9-3)+ 3(4-4)]$$
$$det⁡ A = 1[4(0)-3(16)+ 8(24)]-8[ 2(0)-3(4)+ 8(6)]+ 7[ 2(16)-4(4)+ 8(0)]-2[2(24)-4(6)+ 3(0)]$$
$$det A = 1[0-48+192]-8[0-12+48]+ 7[ 32-16+0]-2[48-24+0]$$
$$det⁡ A = 1-8+ 7-2$$
$$det A = 144-288+112- 48$$
$$det⁡ A = -80$$

#### Leibniz Formula:

For the calculations of matrix A = (aij)4×4 by using Leibniz formula is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c & d\\e & f & g &h \\i & j & k & l \\ m & n & o & p \end{vmatrix} \\$$

$$det A = a*f*k*p + a*j*o*h + a*n*g*l + e*b*o*l + e*j*c*p + e*n*k*d + i*b*g*p + i*f*o*d + i*n*c*h+ m*b*k*h + m*f*c*l + m*j*g*d − a*f*o*l – a*j*g*p – a*n*k*h − e*b*k*p – e*j*o*d -e*n*c*l− i*b*o*h – i*f*c*p – i*n*g*d − m*b*g*l – m*f*k*d – m*j*c*h$$
Example:
Find $$det A = \begin{vmatrix} 1 & 8 & 7 & 2\\2 & 4 & 3 &8 \\1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{vmatrix} \\$$?

Solution:

$$det A = \begin{vmatrix} 1 & 8 & 7 & 2\\2 & 4 & 3 &8 \\1 & 4 & 3 & 2 \\ 1 & 4 & 9 & 6 \end{vmatrix} \\$$
$$1*4*3*6-1*4*2*9-1*3*4*6+1*3*2*4+1*8*4*9-1*8*3*4-8*2*3*6+8*2*2*9+8*3*1*6-8*3*2*1-8*8*1*9+8*8*3*1+7*2*4*6-7*2*2*4-7*4*1*6+7*4*2*1+7*8*1*4-7*8*4*1-2*2*4*9+2*2*3*4+2*4*1*9-2*4*3*1-2*3*1*4+2*3*4*1$$
$$=-80$$

### For 5×5 Matrix Multiplications:

The calculations for 5×5 matrixes from different methods are discussed here:

#### Expand Along the Column:

For the calculations of matrix A = (aij)5×5 from expansion of column is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c & d & e\\f & g & h & i & j\\k & l & m & n & o \\ p & q & r & s & t \\ u & v & w & x & y \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} g & h & i & j\\l & m & n & o\\q & r & s & t\\v & w & x & y\end{vmatrix} – f\begin{vmatrix}b & c & d & e\\l & m & n & o\\ q & r & s & t\\ v & w & x & y\end{vmatrix}+k\begin{vmatrix}b & c & d & e \\g & h & i & j\\q & r & s & t\\v & w & x & y\end{vmatrix}-p\begin{vmatrix}b & c & d & e\\g & h & i & j\\l & m & n & o\\q & r & s & t\end {vmatrix}$$

Then, simply determine the determinant of 4×4 by using above formula of 4×4.

#### Expand Along the Row:

For the calculations of matrix A = (aij)5×5 from expansion of row is determined by the following formula:

$$det A = \begin{vmatrix} a & b & c & d & e\\f & g & h & i & j\\k & l & m & n & o \\ p & q & r & s & t \\ u & v & w & x & y \end{vmatrix} \\$$

$$det⁡ A= a\begin{vmatrix} g & h & i & j\\l & m & n & o\\q & r & s & t\\v & w & x & y\end{vmatrix} – b\begin{vmatrix}g & h & i & j\\k & m & n & o\\ p & r & s & t\\ u & w & x & y\end{vmatrix}+c\begin{vmatrix}f & g & i & j \\k & l & n & o\\p & q & s & t\\u & v & x & y\end{vmatrix}-d\begin{vmatrix}f & g & h & j\\k & l & m & o\\p & q & r & t\\u & v & w & y\end {vmatrix}+e\begin{vmatrix}f & g & h & i\\k & l & m & n\\p & q & r & s\\u & v & w & x\end {vmatrix}$$

Then, simply determine the determinant of 4×4 by using above formula of 4×4

#### Leibniz Formula:

For the calculations of matrix A = (aij)5×5 by using Leibniz formula is determined by the following formula:

$$det A = \begin{vmatrix} a11 & a12 & a13 & a14 & a15\\a21 & a22 & a23 & a24 & a25\\a31 & a32 & a33 & a34 & a35 \\ a41 & a42 & a43 & a44 & a45 \\ a51 & a52 & a53 & a54 & a55 \end{vmatrix} \\$$

Image

Example:
Find $$det A = \begin{vmatrix} 1 & 8 & 7 & 2 & 8\\2 & 4 & 3 &8 & 3\\1 & 4 & 3 & 2 &1\\ 1 & 4 & 9 & 6 & 2 \\ 1 & 5 & 7 & 3 & 4 \end{vmatrix} \\$$?
Solution:
$$det A = \begin{vmatrix} 1 & 8 & 7 & 2 & 8\\2 & 4 & 3 &8 & 3\\1 & 4 & 3 & 2 &1\\ 1 & 4 & 9 & 6 & 2 \\ 1 & 5 & 7 & 3 & 4 \end{vmatrix} \\$$
$$=1*4*3*6*4-1*4*3*2*3-1*4*2*9*4+1*4*2*2*7+1*4*1*9*3-1*4*1*6*7-1*3*4*6*4+1*3*4*2*3+1*3*2*4*4-1*3*2*2*5-1*3*1*4*3+1*3*1*6*5+1*8*4*9*4-1*8*4*2*7-1*8*3*4*4+1*8*3*2*5+1*8*1*4*7-1*8*1*9*5-1*3*4*9*3+1*3*4*6*7+1*3*3*4*3-1*3*3*6*5-1*3*2*4*7+1*3*2*9*5-8*2*3*6*4+8*2*3*2*3+8*2*2*9*4-8*2*2*2*7-8*2*1*9*3+8*2*1*6*7+8*3*1*6*4-8*3*1*2*3-8*3*2*1*4+8*3*2*2*1+8*3*1*1*3-8*3*1*6*1-8*8*1*9*4+8*8*1*2*7+8*8*3*1*4-8*8*3*2*1-8*8*1*1*7+8*8*1*9*1+8*3*1*9*3-8*3*1*6*7-8*3*3*1*3+8*3*3*6*1+8*3*2*1*7-8*3*2*9*1+7*2*4*6*4-7*2*4*2*3-7*2*2*4*4+7*2*2*2*5+7*2*1*4*3-7*2*1*6*5-7*4*1*6*4+7*4*1*2*3+7*4*2*1*4-7*4*2*2*1-7*4*1*1*3+7*4*1*6*1+7*8*1*4*4-7*8*1*2*5-7*8*4*1*4+7*8*4*2*1+7*8*1*1*5-7*8*1*4*1-7*3*1*4*3+7*3*1*6*5+7*3*4*1*3-7*3*4*6*1-7*3*2*1*5+7*3*2*4*1-2*2*4*9*4+2*2*4*2*7+2*2*3*4*4-2*2*3*2*5-2*2*1*4*7+2*2*1*9*5+2*4*1*9*4-2*4*1*2*7-2*4*3*1*4+2*4*3*2*1+2*4*1*1*7-2*4*1*9*1-2*3*1*4*4+2*3*1*2*5+2*3*4*1*4-2*3*4*2*1-2*3*1*1*5+2*3*1*4*1+2*3*1*4*7-2*3*1*9*5-2*3*4*1*7+2*3*4*9*1+2*3*3*1*5-2*3*3*4*1+8*2*4*9*3-8*2*4*6*7-8*2*3*4*3+8*2*3*6*5+8*2*2*4*7-8*2*2*9*5-8*4*1*9*3+8*4*1*6*7+8*4*3*1*3-8*4*3*6*1-8*4*2*1*7+8*4*2*9*1+8*3*1*4*3-8*3*1*6*5-8*3*4*1*3+8*3*4*6*1+8*3*2*1*5-8*3*2*4*1-8*8*1*4*7+8*8*1*9*5+8*8*4*1*7-8*8*4*9*1-8*8*3*1*5+8*8*3*4*1$$
$$=-248$$

Note:

The Triangle rule & Rule of Sarrus only applicable to the matrix till 3×3. Our online matrix determinant calculator uses these all formulas for the accurate & exact calculations of determinants. Simply, you can use our online math calculator that helps you to perform different mathematical operations easily in a fraction of time.

## How to Use This Online Matrix Determinant Calculator:

Our online calculator helps to find the determinant of the matrix up to 5×5 with five different methods. Just follow the points for the accurate outcomes.

Inputs:

• First of all, select the order of the matrix from the dropdown of calculator.
• Then, enter the values of matrix in the designated fields.
• Then, choose the method from which you find the determinant.
• Lastly, hit the calculate button.

Note:

There is a field of ‘column or row number’ in which you enter the row number or column number which you have to expand. Also, there are fields of generate matrix & clear matrix in it, It will automatically generate the matrix & clear all the values from matrix respectively.

Outputs:

Once you fill all the fields, the calculator shows:

• Determinant of the matrix.
• Step-by-Step calculations.

Note:

No matter, which method you select for calculations, the online determinant calculator shows you the results according to the selected option.

## Determinant Properties:

As the determinants have many properties that are useful, but here we listed some of its important properties:

• The determinant of product of numbers is equal to the product of determinants of numbers.
• If we exchange the two rows & two columns of the matrix, then the determinant remains same but with opposite sign.
• A matrix determinant is equal to the transpose of the matrix.
• The determinant of the 5×5 matrix is useful in the Laplace Expansion.
• If we add the same two copies of the first row into any row (columns into any column), then the determinant will not be changed.

### What are determinants used for?

The determinant is helpful in determining the solution of linear equations, capturing how the linear transformation change volume or area & change variables in integrals. It is display as a function whose input is square matrix but output is as single number.

### What does a determinant of 0 mean?

The determinant of 0 means the volume is zero (0). It can only be happen when one of the vector overlap one of the other.

### Can a determinant be negative?

As it is a real number, not a matrix. So, it can be negative number. The determinant only exist for square matrices (2×2, 3×3, … n×n).

## End-Note:

Thankfully, you come to know about the determinants, how to find it manually, and different applications in mathematics including solution of linear equations; determine the change of volume or area in linear transformation etc. When it comes to solve the determinant for higher order matrix, it is very daunting task. Simply, give a try to this online determinant calculator that allows you to find the determinant of the matrices with different calculation methods with complete calculations. Typically, students & professionals use this matrix determinant calculator to solve their mathematical problems.

## References :

From the source of Wikipedia : Definition of determinant & applications

From the site of Semath : Determinant of 5 by 5

From Wikipedia : The rule of Sarrus & its calculations

From tex.stackexchange.com : Triangle rule for determinant