Determinants

Definitions and Formulas

Introduction:- Every square matrix associates with a number that is called the determinant of that square matrix.

For ex:- If A is a square matrix then its determinant is |A| and det(A).

⇒ |A| is not modulus, it is a determinant of A.

⇒ Only a square matrix has a determinant.

Determinant of a matrix of order one:- Let A=[a] be the matrix of order 1 then its determinant is defined as |A| = a.

For ex:- A = [3] then |A| = 3.

Determinant of a matrix of order two:- 

Determinant of a matrix of order three(3x3):- Determinant of a matrix order three can be solved by expressing it into second-order matrices. This is known as expansion of determinant along with a row or a column. So, one matrix is solved by six ways(by three rows or by three columns)

Let square matrix A = [aij]3x3

Expansion of this determinant along the first row(R1):-

Shortcut:- Use these signs instead of the use of (-1)i+j

Properties of Determinants:- These properties are true for determinants of any order.

1. The value of the determinants remains unchanged if its rows and columns are interchanged.

Proof:- Taking L.H.S.

Taking R.H.S.

So, L.H.S. = R.H.S.  (Hence Proved………)

Hence the determinant of matrix A is equal to the determinant of matrix transpose of A.

det(A) = det(A’)

NOTE:- Interchange of rows by Ri ↔ Rj 

Interchange of columns by Ci ↔ Cj 

Interchange of row and column by Ri ↔ Ci 

2. If any two rows (or columns) of a determinant are interchanged, then the sign of determinant changes.

Proof:- Taking L.H.S.

Taking R.H.S.

So, L.H.S. = R.H.S.

3. If any two rows (or columns) of a determinant are identical (all corresponding elements are the same), then the value of the determinant is zero.

4. If each element of the row (or column) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

Proof:-

5. Value of determinant remains same if apply these operations 

Ri → Ri+kRj  or Ci → Ci+kCj .

6. 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. 

Area of Triangle:- If triangle has vertices (x1 , y1), (x2 , y2) and (x3 , y3) is given by this expression:- 1/2[x1(y2 - y3)+x2(y3 - y1)+x3(y1 - y2)]. This expression is written as:-

NOTE:- If the area of the triangle is negative then the area of the triangle is the modulus of that value because the area should be always positive.

⇒ The area of the triangle formed by three collinear points is zero.

Minors:- It is the compact form of the determinant. Minor of an element aij of a determinant obtained by deleting its ith row and jth column in which element aij lies. Minor of an element aij is denoted by Mij.

NOTE:- Minor of an element of a determinant of order n is a determinant of order n-1. (If n≥2)

Cofactors:- Cofactor of an element aij, denoted by Aij is defined by

⇒ Aij = (-1)i+jMij , where Mij is minor of aij.

Hence, the Sum of the product of elements of any rows(or columns) with their corresponding cofactors is defined as a determinant of that matrix.

NOTE:- If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.

⇒ a11A21+a12A22+a13A23 = 0

Adjoint of a Matrix:- The adjoint of a square matrix A=[aij]nxn is defined as the transpose of the cofactor matrix [Aij]nxn, where Aij is the cofactor of the elements aij. Adjoint of matrix A is denoted by adj A.

Theorem 1:- If A be any given square matrix of order n, then

A(adj A) = (adj A)A = |A|xI.

Proof:- Let A is a square matrix of order 3

We know, (i) If elements of a row (or column) are multiplied with cofactors of any other row (or column), then their sum is zero.

(ii) The sum of the product of elements of any rows(or columns) with their corresponding cofactors is defined as a determinant of that matrix.

NOTE:- A square matrix A is said to be singular if |A| = 0.

A square matrix A is said to be non-singular if |A| ≠ 0.

Theorem 2:- If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

Theorem 3:- The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A||B|, where A and B are square matrices of the same order.

NOTE:- We know that (adj A)A = |A|xI

⇒ If A is a nonsingular matrix of order n, then |adj (A)| = |A|n-1.

Theorem 4:- A square matrix A is an invertible matrix if and only if A is a nonsingular matrix.

Proof:- We know that A(adj A) = |A| I ……………(1)

Eq. 1 is Pre-multiplied by A-1:-

⇒ A-1A(adj A) = |A| A-1 I

We know A-1A=I and A-1 I=A-1

⇒ I(adj A) =|A| A-1

Consistent system:- A system of equations is said to be consistent if its solution (one or more) exists. 

Inconsistent system:- A system of equations is said to be inconsistent if its solution does not exist.

Solution of system of linear equations using inverse of a matrix:- System of linear equations is considered as matrix equations and solve them using inverse of the matrix.

Consider the system of equations

a1x+b1y+c1z = d1   ;  a2x+b2y+c2z = d2   ;    a3x+b3y+c3z = d3

Case 1:- If A is a nonsingular matrix, then its inverse exists.

⇒ AX = B

Premultiply by A-1

⇒ A-1AX = A-1B    (A-1A = I)

⇒ IX = A-1B

⇒ X = A-1B

This matrix equation provides a unique solution for the given system of equations. This method of solving systems of equations is known as the Matrix Method.

Case 2:- If A is a singular matrix, then |A| = 0. In this case, we calculate (adj A) B. 

If (adj A) B ≠ O, (O being zero matrix), then the solution does not exist and the system of equations is called inconsistent. 

If (adj A) B = O, then the system may be either consistent or inconsistent as the system has either infinitely many solutions or no solution.

⇒ For a square matrix A in matrix equation AX = B 

(i) |A| ≠ 0, there exists unique solution 

(ii) |A| = 0 and (adj A) B ≠ 0, then there exists no solution 

(iii) |A| = 0 and (adj A) B = 0, then the system may or may not be consistent.

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