Determinant of a matrix

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Determinant of a matrix - definition
Determinant of a matrix - properties
Determinant of a matrix - methods of calculation

The determinant of a matrix is one of the main numerical characteristics of a square matrix, used in solving of many problems.

Definition.

Determinant of a matrix A n×n is number:

det(A) =	Σ	(-1)^{N(α₁,α₂,...,α_n)}·a_α₁1·a_α₂2·...·a_{α_nn}
	(α₁,α₂,...,α_n)

where (α₁,α₂,...,α_n) - permutation of numbers from 1 to n, N(α₁,α₂,...,α_n) - the number of inversions in the permutation, summation goes over all possible permutations of order n.

Notation

The determinant of a matrix A is usually denoted: det(A), |A| or ∆(A).

Determinant of a matrix - properties

The determinant of a identity matrix is equal to one:
det(I_n) = 1
The determinant of a matrix with two equal rows (columns) is equal to zero.
The determinant of a matrix with two proportional rows (columns) is equal to zero.
The determinant of a matrix with a zero row (column) is equal to zero.
The determinant of a matrix is equal to zero if the two or more rows (columns) of this matrix are linearly dependent.
The determinant of a transposed matrix:
det(A) = det(A^T)
The determinant of a inverse matrix:
det(A^-1) = 1det(A)
The determinant of a matrix does not change, if to some of its row (column) to add another row (column) multiplied by some number.
The determinant of a matrix does not change, if to some of its row (column) to add a linear combination of other rows (columns).
If you interchange two rows (columns) of the matrix, the determinant of the matrix changes sign.
The common factor in a row (column) may be taken outside of the determinant:

a₁₁ a₁₂ ... a_1n

a₂₁ a₂₂ ... a_2n

. . . .

k·a_i1 k·a_i2 ... k·a_in

. . . .

a_n1 a_n2 ... a_nn

= k

a₁₁ a₁₂ ... a_1n

a₂₁ a₂₂ ... a_2n

. . . .

a_i1 a_i2 ... a_in

. . . .

a_n1 a_n2 ... a_nn
det(kA) = kⁿ det(A) where A is n×n matrix.

If every element in some row of the determinant is equal to the sum of two terms, the original determinant is equal to the sum of two determinants:

a₁₁	a₁₂	...	a_1n
a₂₁	a₂₂	...	a_2n
.	.	.	.
b_i1 + c_i1	b_i2 + c_i2	...	b_in + c_in
.	.	.	.
a_n1	a_n2	...	a_nn

a₁₁	a₁₂	...	a_1n
a₂₁	a₂₂	...	a_2n
.	.	.	.
b_i1	b_i2	...	b_in
.	.	.	.
a_n1	a_n2	...	a_nn

a₁₁	a₁₂	...	a_1n
a₂₁	a₂₂	...	a_2n
.	.	.	.
c_i1	c_i2	...	c_in
.	.	.	.
a_n1	a_n2	...	a_nn

The determinant of the upper triangular matrix equal to the product of its diagonal elements.
The determinant of the matrix product equal to the product the determinants of these matrices:
det(A·B) = det(A)·det(B)

Determinant of a matrix - methods of calculation

Determinant of 1×1 matrix

Rule:

For the matrix of the first order the value of determinant equal to the value matrix element:

∆ = |a₁₁| = a₁₁

Determinant of 2×2 matrix

Rule:

For a matrix of 2×2 the determinant is equal to the difference between the value of products of elements of the main diagonal and antidiagonal:

∆ =

a₁₁	a₁₂
a₂₁	a₂₂

= a₁₁·a₂₂ - a₁₂·a₂₁

Example 1.

Find determinant of a matrix A

A =

	5	7
	-4	1

Solution:

det(A) =

5	7
-4	1

= 5·1 - 7·(-4) = 5 + 28 = 33

Determinant of 3×3 matrix

Triangle's rule

Rule:

The value of the determinant is equal to the sum of products of main diagonal elements and products of elements lying on the triangles with side which parallel to the main diagonal, from which subtracted the product of the antidiagonal elements and products of elements lying on the triangles with side which parallel to the antidiagonal.


+		–

∆ =

a₁₁	a₁₂	a₁₃
a₂₁	a₂₂	a₂₃
a₃₁	a₃₂	a₃₃

= a₁₁·a₂₂·a₃₃ + a₁₂·a₂₃·a₃₁ + a₁₃·a₂₁·a₃₂ - a₁₃·a₂₂·a₃₁ - a₁₁·a₂₃·a₃₂ - a₁₂·a₂₁·a₃₃

Sarrus' rule

Rule:

Write out the first 2 columns of the matrix to the right of the 3rd column, so that you have 5 columns in a row. Then add the products of the diagonals going from top to bottom and subtract the products of the diagonals going from bottom to top:

∆ =

a₁₁	a₁₂	a₁₃	a₁₁	a₁₂
a₂₁	a₂₂	a₂₃	a₂₁	a₂₂
a₃₁	a₃₂	a₃₃	a₃₁	a₃₂

= a₁₁·a₂₂·a₃₃ + a₁₂·a₂₃·a₃₁ + a₁₃·a₂₁·a₃₂ - a₁₃·a₂₂·a₃₁ - a₁₁·a₂₃·a₃₂ - a₁₂·a₂₁·a₃₃

Example 2.

Find determinant of a matrix A

A =

5	7	1
-4	1	0
2	0	3

Solution:

det(A) =

5	7	1
-4	1	0
2	0	3

= 5·1·3 + 7·0·2 + 1·(-4)·0 - 1·1·2 - 5·0·0 - 7·(-4)·3 =

= 15 + 0 + 0 - 2 - 0 + 84 = 97

Determinant of n×n matrix

Expanding to Find the Determinant

Rule: Expanding by row

Pick any row in the matrix. It does not matter which row you use, the answer will be the same for any row.
Multiply every element in that row by its cofactor and add. The result is the determinant:

	n
det(A) =	Σ	a_ij·A_ij - expanding by i row
	j = 1

Rule: Expanding by column

Pick any column in the matrix. It does not matter which column you use, the answer will be the same for any column.
Multiply every element in that column by its cofactor and add. The result is the determinant:

	n
det(A) =	Σ	a_ij·A_ij - expanding by j column
	i = 1

Some rows or columns are better than others:

Pick the row or column with the most zeros in it. (Since each minor or cofactor is multiplied by the zero element is equal zero)
Pick the row or column with the largest numbers (or variables) in it. (The elements in the row or column that you expand along are not used to find the minors. Is easier calculate cofactors with smallest numbers)

Example 3.

Find determinant of a matrix A

A =

2	4	1
0	2	1
2	1	1

Solution: Expand determinant on the first column:

det(A) =

2	4	1
0	2	1
2	1	1

= 2·(-1)¹⁺¹·

2	1
1	1

+ 0·(-1)²⁺¹·

4	1
1	1

+ 2·(-1)³⁺¹·

4	1
2	1

= 2·(2·1 - 1·1) + 2·(4·1 - 2·1) = 2·(2 - 1) + 2·(4 - 2) = 2·1 + 2·2 = 2 + 4 = 6

Example 4.

Find determinant of a matrix A

A =

2	4	1	1
0	2	0	0
2	1	1	3
4	0	2	3

Solution: Expand determinant on the second row:

det(A) =

2	4	1	1
0	2	0	0
2	1	1	3
4	0	2	3

= -0·

4	1	1
1	1	3
0	2	3

+ 2·

2	1	1
2	1	3
4	2	3

- 0·

2	4	1
2	1	3
4	0	3

+ 0·

2	4	1
2	1	1
4	0	2

= 2·(2·1·3 + 1·3·4 + 1·2·2 - 1·1·4 - 2·3·2 - 1·2·3) = 2·(6 +12 + 4 - 4 - 12 - 6) = 2·0 = 0

Transform matrix to upper triangular form

Rule:

Using the properties of the determinant 8 - 11 for elementary row and column operations transform matrix to upper triangular form. Determinant of of the upper triangular matrix equal to the product of its main diagonal elements.

Example 5.

Find determinant of a matrix A

A =

2	4	1	1
0	2	1	0
2	1	1	3
4	0	2	3

Solution: transform matrix to upper triangular form

det(A) =

2	4	1	1
0	2	1	0
2	1	1	3
4	0	2	3

R₃ - R₁ → R₃ (multiply 1 row by -1 and add it to 3 row); R₄ + 2R₁ → R₄ (multiply 1 row by 2 and add it to 4 row):

det(A) =

2	4	1	1
0	2	1	0
2 - 2	1 - 4	1 - 1	3 - 1
4 - 2·2	0 - 2·4	2 - 2·1	3 - 2·1

2	4	1	1
0	2	1	0
0	-3	0	2
0	-8	0	1

C₂ ↔ C₃ (interchange the 2 and 3 lolumns):

det(A) = -

2	1	4	1
0	1	2	0
0	0	-3	2
0	0	-8	1

C₃ + 8C₄ → C₃ (multiply 4 column by 8 and add it to 3 column):

det(A) = -

2	1	4 + 8·1	1
0	1	2 + 8·0	0
0	0	-3 + 8·2	2
0	0	-8 + 8·1	1

= -

2	1	12	1
0	1	2	0
0	0	13	2
0	0	0	1

= -2·1·13·1 = -26

Online calculators with matrixes

Online exercises with matrixes

Matrix Matrix Definition. Main information System of linear equations - matrix form Types of matrices Matrix scalar multiplication Addition and subtraction of matrices Matrix multiplication Transpose matrix Elementary matrix operations Determinant of a matrix Minors and cofactors of a matrix Inverse matrix Linearly dependent and independent rows Rank of a matrix

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