Chapter 4
DETERMINANTS
4.1 Overview
To every square matrix A = [aij] of order n, we can associate a number (real or complex)
called determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.
a b
If A = , then determinant of A, denoted by |A| (or det A), is given by
c d
a b
|A| = = ad – bc.
c d
Remarks
(i) Only square matrices have determinants.
(ii) For a matrix A, A is read as determinant of A and not, as modulus of A.
4.1.1 Determinant of a matrix of order one
Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a.
4.1.2 Determinant of a matrix of order two
a b
Let A = [aij] = be a matrix of order 2. Then the determinant of A is defined
c d
as: det (A) = |A| = ad – bc.
4.1.3 Determinant of a matrix of order three
The determinant of a matrix of order three can be determined by expressing it in terms
of second order determinants which is known as expansion of a determinant along a
row (or a column). There are six ways of expanding a determinant of order 3
corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and
C3) and each way gives the same value.
, 66 MATHEMATICS
Consider the determinant of a square matrix A = [aij]3×3, i.e.,
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
Expanding |A| along C1, we get
a22 a23 a12 a13 a12 a13
|A| = a11 (–1)1+1 a a33 + a21
(–1) 2+1
a32 a33 + a31
(–1) 3+1
a22 a23
32
= a11(a22 a33 – a23 a32) – a21 (a12 a33 – a13 a32) + a31 (a12 a23 – a13 a22)
Remark In general, if A = kB, where A and B are square matrices of order n, then
|A| = kn |B|, n = 1, 2, 3.
4.1.4 Properties of Determinants
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ = transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant
changes.
(iii) If any two rows or any two columns in a determinant are identical (or
proportional), then the value of the determinant is zero.
(iv) Multiplying a determinant by k means multiplying the elements of only one row
(or one column) by k.
(v) If we multiply each element of a row (or a column) of a determinant by constant
k, then value of the determinant is multiplied by k.
(vi) If elements of a row (or a column) in a determinant can be expressed as the
sum of two or more elements, then the given determinant can be expressed as
the sum of two or more determinants.
, DETERMINANTS 67
(vii) If to each element of a row (or a column) of a determinant the equimultiples of
corresponding elements of other rows (columns) are added, then value of
determinant remains same.
Notes:
(i) If all the elements of a row (or column) are zeros, then the value of the determinant
is zero.
(ii) If value of determinant ‘∆’ becomes zero by substituting x = α, then x – α is a
factor of ‘∆’.
(iii) If all the elements of a determinant above or below the main diagonal consists of
zeros, then the value of the determinant is equal to the product of diagonal
elements.
4.1.5 Area of a triangle
Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by
x1 y1 1
1
∆ = x2 y2 1
2 .
x3 y3 1
4.1.6 Minors and co-factors
(i) Minor of an element aij of the determinant of matrix A is the determinant obtained
by deleting ith row and jth column, and it is denoted by Mij.
(ii) Co-factor of an element aij is given by Aij = (–1)i+j Mij.
(iii) Value of determinant of a matrix A is obtained by the sum of products of elements
of a row (or a column) with corresponding co-factors. For example
|A| = a11 A11 + a12 A12 + a13 A13.
(iv) If elements of a row (or column) are multiplied with co-factors of elements of
any other row (or column), then their sum is zero. For example,
a11 A21 + a12 A22 + a13 A23 = 0.
4.1.7 Adjoint and inverse of a matrix
(i) The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix
DETERMINANTS
4.1 Overview
To every square matrix A = [aij] of order n, we can associate a number (real or complex)
called determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.
a b
If A = , then determinant of A, denoted by |A| (or det A), is given by
c d
a b
|A| = = ad – bc.
c d
Remarks
(i) Only square matrices have determinants.
(ii) For a matrix A, A is read as determinant of A and not, as modulus of A.
4.1.1 Determinant of a matrix of order one
Let A = [a] be the matrix of order 1, then determinant of A is defined to be equal to a.
4.1.2 Determinant of a matrix of order two
a b
Let A = [aij] = be a matrix of order 2. Then the determinant of A is defined
c d
as: det (A) = |A| = ad – bc.
4.1.3 Determinant of a matrix of order three
The determinant of a matrix of order three can be determined by expressing it in terms
of second order determinants which is known as expansion of a determinant along a
row (or a column). There are six ways of expanding a determinant of order 3
corresponding to each of three rows (R1, R2 and R3) and three columns (C1, C2 and
C3) and each way gives the same value.
, 66 MATHEMATICS
Consider the determinant of a square matrix A = [aij]3×3, i.e.,
a11 a12 a13
A = a21 a22 a23
a31 a32 a33
Expanding |A| along C1, we get
a22 a23 a12 a13 a12 a13
|A| = a11 (–1)1+1 a a33 + a21
(–1) 2+1
a32 a33 + a31
(–1) 3+1
a22 a23
32
= a11(a22 a33 – a23 a32) – a21 (a12 a33 – a13 a32) + a31 (a12 a23 – a13 a22)
Remark In general, if A = kB, where A and B are square matrices of order n, then
|A| = kn |B|, n = 1, 2, 3.
4.1.4 Properties of Determinants
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ = transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant
changes.
(iii) If any two rows or any two columns in a determinant are identical (or
proportional), then the value of the determinant is zero.
(iv) Multiplying a determinant by k means multiplying the elements of only one row
(or one column) by k.
(v) If we multiply each element of a row (or a column) of a determinant by constant
k, then value of the determinant is multiplied by k.
(vi) If elements of a row (or a column) in a determinant can be expressed as the
sum of two or more elements, then the given determinant can be expressed as
the sum of two or more determinants.
, DETERMINANTS 67
(vii) If to each element of a row (or a column) of a determinant the equimultiples of
corresponding elements of other rows (columns) are added, then value of
determinant remains same.
Notes:
(i) If all the elements of a row (or column) are zeros, then the value of the determinant
is zero.
(ii) If value of determinant ‘∆’ becomes zero by substituting x = α, then x – α is a
factor of ‘∆’.
(iii) If all the elements of a determinant above or below the main diagonal consists of
zeros, then the value of the determinant is equal to the product of diagonal
elements.
4.1.5 Area of a triangle
Area of a triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is given by
x1 y1 1
1
∆ = x2 y2 1
2 .
x3 y3 1
4.1.6 Minors and co-factors
(i) Minor of an element aij of the determinant of matrix A is the determinant obtained
by deleting ith row and jth column, and it is denoted by Mij.
(ii) Co-factor of an element aij is given by Aij = (–1)i+j Mij.
(iii) Value of determinant of a matrix A is obtained by the sum of products of elements
of a row (or a column) with corresponding co-factors. For example
|A| = a11 A11 + a12 A12 + a13 A13.
(iv) If elements of a row (or column) are multiplied with co-factors of elements of
any other row (or column), then their sum is zero. For example,
a11 A21 + a12 A22 + a13 A23 = 0.
4.1.7 Adjoint and inverse of a matrix
(i) The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix
