Determinants

Determinants

With every square matrix A, we associate a number, called the determinant of A and denoted by |A| or det (A). A determinant of order n is defined as

|A| = where a_ij is the general element.

Consider the equations a₁x + b₁y = 0, a₂x + b₂y = 0. These give

a₁b₂ - a₂b₁ = 0.

We express this eliminant as = 0.



A determinant of order three consisting of 3 rows and 3 columns is written as and is equal to

.

The numbers a_i, b_i, c_i ( i =1,2,3 ) are called the elements of the determinant.

The determinant, obtained by deleting the ith row and the jth column is called the minor of the element at ith row and jth column i.e. a_ij. This new determinant is of order (n - 1), and is denoted by M_ij. The cofactor of this element a_ij is (-1)^i+j (minor). i.e. A_ij = (-1)^i+j M_ij. In the third order determinant = = a₁A₁ + b₁B₁+c₁C₁,

A₁, B₁ and C₁ are the cofactors of a₁, b₁ and c₁ respectively, and

A₁ = .

We can expand a determinant through any row or column. It means we can write





These results are true for determinants of any order.


Properties of Determinants

The following properties are true for determinants of all orders. However, we will provide proofs only for a third order determinant.

(i) If rows be changed into columns and columns into the rows, then the values of the determinant remains unaltered, (i.e. if the corresponding rows and columns

are interchanged) or |A| = |A'|.

Since the determinant can be expanded through any row or any column, the value of the determinant remains unchanged even if the rows and columns are

interchanged.

(ii) If any two rows (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.

(iii) If all the elements of a row (or a column) are zero, then the value of the determinant is zero.

(iv) If two rows (or two columns) in a determinant have corresponding elements that are equal, the value of the determinant is equal to zero.

(v) If each of the elements of one row (or columns) of a determinant is multiplied by k, then the new determinant is k times the original determinant.

A common factor of any row or any column of the determinant can be taken out of the determinant as a factor.

Note: Let A = (a_ij) be a square matrix of order n. Then kA = (ka_ij) |kA| = kⁿ |A|

since the factor k is taken out of the determinant from each of the n rows or n columns.

(vi) If the corresponding elements of any two rows (or columns) of a determinant are proportional, the value of the determinant is zero.

(vii) If each element in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or

more determinants.

(viii) If to each element of a line (row or column) of a determinant be added the equimutiples of the corresponding elements of one or more parallel lines, the

determinant remains unaltered i.e.




=





= = .


(ix) If a determinant D vanishes for x = a, then (x - a) is a factor of D, In other words, if two rows (or two columns) become identical for x = a then (x - a) is a

factor of D.

In general, if r rows (or r columns) become identical when a is substituted for x, then (x - a)^r-1 is a factor of D.


Product of Two Determinants

We can write




Here we have multiplied rows by rows. We can also multiply rows by columns or columns by rows, or columns by columns. It follows from the fact that |A| = |A'|.

Moreover, |AB| = |A| |B| = |B| |A| where A and B are square matrices of the same order.

Note: If = |a_ij| is a determinant of order n, then the value of the determinant |A_ij|, where A_ij is the cofactor of a_ij, is ^n-1. This is known as power cofactor

formula.

Special Determinants 1.

Symmetric determinant

The elements situated at equal distance from the diagonal are equal both in magnitude and sign, i.e. |a_ij| = |a_ji|.

2. Skew symmetric determinant

All the diagonal elements are zero and the elements situated at equal distance from the diagonal are equal in magnitude but opposite in sign. The value of a skew

symmetric determinant of odd order is zero.

3. Circulant determinant:

The elements of the rows (or columns) are in cyclic arrangement.
4. .


5. .


6. .


Singular and Non-singular Matrix:

A square matrix A is said to be non-singular if 0, and a square matrix A is said to be singular if = 0. Here |A| (or det(A) or simply det A) means

corresponding determinant of the square matrix A:

Unitary Matrix:

A square matrix is said to be unitary if`A = I. Since || = |A| and

|A| = || |A|, || |A| = 1.

Thus the determinant of unitary matrix is of unit modulus. For a matrix to be unitary it must be non-singular.

Orthogonal Matrix:

A square matrix A of order n is said to be orthogonal if AA' = A' A = .


Idempotent Matrix:

A square matrix A is called idempotent if it satisfies the relation A² = A.


Involutary Matrix:

A matrix A, such that A² = I is called an involutary matrix.

Nilpotent Matrix:

A square matrix A is called a nilpotent matrix if there exists a positive integer m such that A^m = O. If m is the least positive integer such that A^m = O, then m is

called the index of the nilpotent matrix A


Adjoint of a Square Matrix

Let A = [a_ij] be a square matrix of order n and let C_ij be the cofactor of a_ij in A. Then the transpose of the matrix of cofactors of elements of A is called the

adjoint of A and is denoted by adj A.

Thus, adjA = [C_ij]^T (adj A)_ij = C_ji,

If A = , then, adjA =

where C_ij denotes the cofactor of a_ij in A.

Theorem: Let A be a square matrix of order n. Then A(adj A) = |A| I_n = (adj A)A

Note:

The adjoint of a square matrix of order 2 can easily be obtained by interchanging the diagonal elements and changing the signs of off-diagonal (left hand

side lower corner to right hand side upper corner) elements.


Inverse of a Matrix

A non-singular square matrix of order n is said to be invertible if there exists a square matrix B of the same order such that AB = I_n = BA.

In such a case, we say that the inverse of A is B and we write, A^-1 = B, or A = B^-1. Hence, A^-1 is the inverse of A if AA^-1 = A^-1A = I
.
We know that A adj (A) = |A| I_n

A^-1A adj (A) = |A| A^-1

I adj (A) = |A| A^-1 adj A = |A| A^-1.

The inverse of A is thus, given by A^-1 =. adj A.

Properties of Inverse of a Matrix

(i). Every invertible matrix possesses a unique inverse.

(ii). (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)^-1 = B^-1A^-1.

(iii).If A is an invertible square matrix, then A' is also invertible and (A')^-1 = (A^-1)'.

(iv).If A is a non-singular square matrix of order n, then |adjA| = |A|^n-1

(v). If A and B are non-singular square matrices of the same order, then

adj (AB) = (adj B) (adj A)

(vi).If A is an invertible square matrix, then adj(A') = (adj A)'

(vii).If A is a non-singular square matrix, then adj(adjA) = |A|^n-2 A

(viii)For any non-singular matrix A and positive integer k, A^-k = (A^k)^-1 = (A^-1)^k.

Elementary Transformations or Elementary Operations on a Matrix:

The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations.

(i). Interchange of any two rows (columns)

If ith row (column) of a matrix is interchanged with the jth row (column), it will be denoted by R_I R_j (C_I C_j).

(ii). Multiplying all elements of a row (column) of a matrix by a non-zero scalar

If the elements of ith row (column) are multiplied by non-zero scalar k, it will be denoted by R_I R_i (k) [C_i C_i (k)] or R_I kR_i [C_i kC_i].

(iii).Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar k.

If k times the elements of j^th row (column) are added to the corresponding elements of the i^th row (column), it will be denoted by R_i R_i +k R_j (C_i C_i + kC_j).

Note: (i) An elementary row operation on the product of two matrices is equivalent to the same elementary row operation on the pre-factor.

(ii) An elementary column operation on the product of two matrices is equivalent to the same elementary column operation on the post-factor.

System of Simultaneous Linear Equations

Consider the following system of n linear equations in n unknowns:

a₁₁ x₁ + a₁₂ x₂ + ............+ a_1n x_n = b₁

a₂₁ x₁ + a₂₂ x₂ + ............+ a_2n x_n = b₂
. . . .

a_n1 x₁ + a_n2 x₂ + ............+ a_nn x_n = b_n

This system of equations can be written in matrix form as



or AX = B

The n × n matrix A is called the coefficient matrix of the system of linear equations.

Homogeneous and Non-Homogeneous System of Linear Equations

A system of equations AX = B is called a homogeneous system if B = O. Otherwise, it is called a non-homogeneous system of equations.

Solution of a System of Equations

Consider the system of equations AX = B.

A set of values of the variables x₁, x₂, ... , x_n which simultaneously satisfy all the equations is called a solution of the system of equations.

Consistent System

If the system of equations has one or more solutions, then it is said to be a consistent system of equations; otherwise it is an inconsistent system of equations.

Solution of a Non-Homogeneous System of Linear Equations:

There are two methods for solving a non-homogeneous system of simultaneous linear equations.

(i). Cramer's Rule

Consider a system of simultaneous linear equations in three variables namely x, y, z i.e.

a₁x + b₁y + c₁z = d₁, ....(1)

a₂x + b₂y + c₂z = d₂, ....(2)

a₃x + b₃y + c₃z = d₃. ....(3)

The following cases can arise:

(1) D 0: In such a case, the system has precisely one solution (unique solution), which is given by 'CRAMER RULE' i.e. x = D₁/D; y = D₂/D; z = D₃/D.

(2) D = 0 : and at least one of the determinants D₁ ,D₂ or D₃ is non-zero, then the system is inconsistent . i.e. it has no solution.

(3) D =0 and D₁ =D₂= D₃ = 0, then the system has either infinite number of solutions (when the equations are dependent) or they have no solution (when they are

independent).

Homogeneous System:

If d₁ = d₂ = d₃ = 0, then the system is known as a system of homogeneous linear equations, and thus, D₁ =D₂ = D₃ = 0 (value of a determinant is zero, if one

column has all elements = 0). The trivial solution (x = 0, y = 0, z = 0) always exists and for the existence of non trivial solutions (infinite solutions) D = 0. If D 0,

then the system has only the trivial or zero solution. x = 0, y = 0 and z = 0. It means that the system has non-trivial solutions only if the matrix A, of the system AX

= 0, is singular i.e. |A| = 0.

Note: A system of three linear equations in two unknown i.e.

a₁x+b₁y +c₁ =0

a₂x+b₂y +c₂ = 0

a₃x+b₃y +c₃ = 0

is consistent if = 0.