Last Updated on: 10th January 2024, 02:36 pm
Determinant
Determinant of Matrix
Determinant is a scalar value calculated from a square matrix. Determinant encodes certain properties of linear transformation described by the matrix.
The Determinant of a matrix A is denoted det(A), det A, or |A|.
So, if A= I a b I
I c d I
then Determinant of A is denoted by det(A), det A, or |A| (for a Matrix, it is read as Determinant A, and not as modulus A).
Only square matrices have determinants. The number of rows or columns of the square matrix is called the Dimension or Order of the Matrix
Evaluation of Determinant of Matrix
One Dimension Matrix : For one dimension matrix [A], the Determinant is the scalar value A
Two Dimension Matrix
I a11 a12 I
I a21 a22 I
be a square matrix of order 2×2. Then the Determinant of A is defined as a11×a22 – a21×a12
Evaluation of Determinant of 2 dimension Matrix – Example 1 : Evaluate I 2 4 I
I -1 2 I
I 2 4 I=2×(2) – 4×(–1) = 4 + 4 = 8.
I -1 2 I
Three Dimension Matrix
a11 a12 a13
I a21 a22 a23 I
a31 a32 a33
is a 
matrix. The Determinant is computed as A= a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23) + a13 (a 21 a 32 – a31 a22).
Three Dimension Matrix – Steps of Computation
The 3 components may be computed in following steps:
- Step 1: Compute the 1st component of Determinant : a11× I a22 a23 I
I a32 a33 I
= a11(a22 a33 – a32 a23) (multiply first element of first row and the determinant obtained by eliminating the first row and first column). - Step 2:Compute the 2nd component of Determinant : a12× I a21 a23 I
I a31 a33 I
= a12(a21 a33 – a31 a23) (multiply second element of first row and the determinant obtained by eliminating the first row and second column). - Step 3:Compute the 3rd component of Determinant:a13× I a21 a22 I
I a31 a32 I
= a13(a 21 a 32 – a31 a22). (multiply third element of first row and the determinant obtained by eliminating the first row and third column).
Determinant Component Sign
The sign id each component is determined by the expression [(–1)sum of suffixes].
- 1st element is multiplied by -1(1+1)= (-1)2=1. So, the sign of first component positive.
- 2nd element is multiplied by (-1)(1+2)= -(1)3=-1. So, the sign of 2nd element is negative.
- 3rd element is multiplied by -1(1+3)= (-1)4=1. So, the sign of 3rd component positive.
The rule is explained for learning. For quick computation, you may simply put the sign as positive or negative, if sum of components is respectively even or odd.
Now add all the 3 components to get the Determinant.
A= a11 (a22 a33 – a32 a23) – a12 (a21 a33 – a31 a23) + a13 (a 21 a 32 – a31 a22).
Evaluation of Determinant of 3 dimension Matrix – Example 1
Evaluate A= 1 2 4
I -1 3 0 I
4 1 0
We observe that the third column contains maximum number of zeros. So, expanding along third column would be easier. So, we get
4× I -1 3 I – 0× I 1 2 I + 0 × I 1 2 I = 4 ×(-1-12) -0 +0 = 48
I 4 1 I I 4 1 I I -1 3 I
Properties of Determinants
a1 a2 a3 a1 b1 c1
I b1 b2 b3 I = I a2 b2 c2 I
c1 c2 c3 a3 b3 c3
The 1st row [a1 a2 a3] becomes, 1st column a1
I a2 I
a3
The 2nd row [b1 b2 b3] becomes, 2nd column b1
I b2 I
b3
The 3rd row [c1 c2 c3] becomes, 3rd column c1
I c2 I
c3
2. If any two rows or columns of a Determinant are interchanged, then sign of determinant changes.
a1 a2 a3 c1 c2 c3
A= I b1 b2 b3 I B= I b1 b2 b3 I , Then Det (A) = – (Det A).
c1 c2 c3 a1 a2 a3
a1 a2 a3 a3 a2 a1
A= I b1 b2 b3 I C= I b3 b2 b1 I , Then Det (A) = – (Det C).
c1 c2 c3 c3 c2 c1
3. If any two rows (or columns) of a Determinant are identical (all corresponding elements are same), then value of Determinant is zero.
a1 a2 a3 a1 a2 a1
A= I b1 b2 b3 I, B= I b1 b2 b1 I , Then Det (A)=0, Det (B)=0,
a1 a2 a3 c1 c2 c1
4. If each element of a row (or a column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.
a1 a2 a3, k x a1 k x a2 k x a3 k x a1 a2 a3
A= I b1 b2 b3 I , B= I b1 b2 b3 I , C= I k x b1 b2 b3 I ,
c1 c2 c3 c1 c2 c3 k x c1 c2 c3
Then Det (B)=k x Det (A), Det (C)=k x Det (A),
5. If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
