Chapter 4: Determinants

Chapter-4

Determinants

A determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of products. Below is an example of a 3 × 3 determinant (it has 3 rows and 3 columns). The result of multiplying out, then simplifying the elements of a determinant is a single number (a scalar quantity).

Concept of Determinant and Properties of Determinants:

A  system of linear equations like

a₁ x + b₁ y = c₁

a₂ x + b₂ y = c₂

Now, this system of equations has a unique solution or not, is determined by the number.

a1 b2 – a2 b1 ( Recall that if

or, a1 b2 – a2 b1 0, then the system of linear

equations has a unique solution). The number  a1 b2 – a2 b1 which determines uniqueness of solution is associated with the matrix  A = a1a2b1b2   

and is called the determinant of A or det A.

Determinants have wide applications in Engineering, Science, Economics, Social Science, etc.

Determinant :

The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or |A|.

In the case of a 2 × 2 matrix the determinant can be defined as

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where a = (i, j)^th element of A. This may be thought of as a function which associates each square matrix with a unique number (real or complex).

If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A ∈ M and k ∈ K, then f (A) is called the determinant of A. It is also denoted by | A | or det A or Δ.

Properties of Determinants

Let us check the below seven properties of determinant in detail. The working principle and the formulas, explanation of each of the properties is also presented below.

1. Interchange Property

The value of a determinant remains unchanged if the rows or the columns of a determinant are interchanged.

Det(A) = Det(A')

It follows from this property that if the rows and columns of the matrix are interchanged, then the transpose of the matrix is obtained and the determinant value and the determinant of the transpose are equal.

2. Sign Property

The sign of the value of determinant changes if any two rows or any two columns are interchanged.

Det(A) = -Det(B)

The value of the determinant only changes the sign if the row or the column is swapped once. In the above matrix A, the second row has been swapped with the third row to obtain matrix B, and we have Det(A) = -Det(B). If the value of the determinant is D, and the rows or columns are swapped n times, then the new value of the determinant is (-1)ⁿD.

3. Zero Property

The value of a determinant is equal to zero if any two rows or any two columns have the same elements.

Here the elements of the first row and the second row are identical. Hence the value of the determinant is equal to zero.

Det(A) = 0

4. Scalar Multiplication Property

The value of the determining becomes k times the earlier value of the determinant if each of the elements of a particular row or column is multiplied with a constant k.

Det(B) = k× Det(B)

The elements of the first row are multiplied with a constant k, and the determinant value is also multiplied with the constant k. This property helps in taking a common factor from each row or a column of the determinant. Also if the corresponding elements of any two rows or columns are equal then the value of the determinant is equal to zero.

5. Sum Property

If a few elements of a row or column are expressed as a sum of terms, then the determinant can be expressed as a sum of two or more determinants.

The elements of the first row represent the sum of terms, which can be split into two different determinants. Further, the new determinants also have the same second and third row, as the earlier determinant.

6. Property Of Invariance

If each element of a row and column of a determinant is added with the equimultiples of the elements of another row or column of a determinant, then the value of the determinant remains unchanged. This can be expressed in the form of a formula as Ri→Ri+kRj, or Ci→Ci+kCj.

The elements of the first row of matrix A have been replaced with the sum of the elements of the first row, and the third row multiplied with a constant, to obtain the new matrix B. Here, after this operation also, the determinant A is equal to determinant B.

7. Triangular Property

If the elements above or below the main diagonal are equal to zero, then the value of the determinant is equal to the product of the elements of the diagonal matrix.

8. Factor Property:

If a determinant Δ becomes zero when we put

x=α,

Then
(x-α)

is a factor of Δ.

9. Determinant of cofactor matrix:

  where C_ij denotes the cofactor of three element a_ij in Δ . 

  

10. Switching Property:

The interchange of any two rows (or columns) of the determinant changes its sign.

Question : Using properties of determinants, prove that

Solution:

By using invariance and scalar multiple property of determinant we can prove the given problem.

Area of a Triangle

The area of a triangle formed by the vertices - (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by

½[x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)] sq.units

It can be expressed in the form of a determinant.

Note: We always take 1 in the last column of the determinant.

Expanding along the first column, we get

 = x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)

The area of a triangle formed by the vertices - (x₁, y₁), (x₂, y₂) and (x₃, y₃) –

Note:

In calculating the area of a triangle by using this formula, we need to take the absolute value of the above determinant to avoid negative values, if any.

If area is given, use both positive and negative values of the determinant for the calculation.

The area of a triangle formed by three collinear points is equal to zero.

 

Example: Find out the area of the triangle whose vertices are given by A(0,0) , B (3,1) and C (2,4).

Solution: Using determinants we can find out the area of the triangle obtained by joining these points using the formula

Example : If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

Solution:

Area of triangle  = 4 square units 

• (1/2){k [4 – 2] – 2[2 – 3] + 1[4 – 12]} = 4
• k(2) – 2(-1) + 1(-8) = 8
• 2k + 2 – 8 = 8
• 2k – 6 = 8
• 2k = 8 + 6
• 2k = 14      => k = 7

So, the value of k is 7.

• (1/2){k [4 – 2] – 2[2 – 3] + 1[4 – 12]} = -4
• k(2) – 2(-1) + 1(-8) = -8
• 2k + 2 – 8 = -8
• 2k – 6 = -8
• 2k = -8 + 6
• 2k = -2      => k = -1

So, the value of k is -1.

Minors and Cofactors:

Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. e.g. in the determinant

Row and Column Operations :

Adjoint and Inverse of a Matrix

The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. For a matrix A, the adjoint is denoted as adj (A). On the other hand, the inverse of a matrix A is that matrix which when multiplied by the matrix A give an identity matrix.

Note.: For a square matrix of order 2, given by

Let the determinant of a square matrix A be     |A|

Then the transpose of the matrix of co-factors is called the adjoint of the matrix A and is written as adj( A )

The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinant A.

Let A be a square matrix, then (Adjoint A). A = A. (Adjoint A) = | A |. I

Note:

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

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

Theorem : If A is a square matrix of order n, then |adj(A)| = |A|^{n – 1}.

Properties of adjoint matrix:

Let A and B  be two non singular square matrix of same order n,

• Property 1:  A (adj(A)) = (adj(A)) A = |A|I, where I is the identitiy matrix of order n.
• Property 2: A square matrix A is invertible if and only if A is a non-singular matrix.
• Property 3: adj(A’)= (adj (A))’
• Property 4: adj(kA)= k^n-1adj (A) , k Î R
• Property 5: adj(A^m)= (adj (A))^m
• Property 6: adj(adj(A))= |A|^n-2A
• Property 7: |adj(A)|= |A|^n-1
• Property 8: |adj(adj(A))|=|A|^(n-1)2 

Example : If A^T = – A then the elements on the diagonal of the matrix are equal to

(a) 1 (b) -1 (c) 0 (d) none of these

Solution:

(c) A^T = -A; A is skew-symmetric matrix; diagonal elements of A are zeros.

so option (c) is the answer.

Example : If A and B are two skew-symmetric matrices of order n, then,

(a) AB is a skew-symmetric matrix (b) AB is a symmetric matrix

(c) AB is a symmetric matrix if A and B commute (d)None of these

Solution:

(c) We are given A’ = -A and B’ = -B;

Now, (AB)’ = B’A’ = (-B) (-A) = BA = AB, if A and B commute.

Example : Let A and B be two matrices such that AB’ + BA’ = 0. If A is skew symmetric ,then BA

(a) Symmetric (b) Skew symmetric (c) Invertible (d) None of these

Solution:

(c) we have, (BA)’ = A’B’ = -AB’ [ A is skew symmetric]; = BA’ = B(-A)

= -BA

BA is skew symmetric.

Example : Which of the following statements are false –

(a) If | A | = 0, then | adj A | = 0;

(b) Adjoint of a diagonal matrix of order 3 × 3 is a diagonal matrix;

(c) Product of two upper triangular matrices is an upper triangular matrix;

(d) adj (AB) = adj (A) adj (B);

Solution:

(d) We have, adj (AB) = adj (B) adj (A) and not adj (AB) = adj (A) adj (B)

Inverse of a Matrix:

If A and B are two square matrices of the same order, such that AB = BA = I (I = unit matrix),Then B is called the inverse of A,

 i.e. B = A^–1 and A is the inverse of B.

Condition for a square matrix A to possess an inverse is that the matrix A is non-singular, i.e., | A | ≠ 0. If A is a square matrix and B is its inverse then AB = I.

Taking determinant of both sides | AB | = | I | or | A | | B | = I.

From this relation it is clear that | A | ≠ 0, i.e. the matrix A is non-singular.

We know that,

Properties of Inverse matrix:

Let A and B  be two non singular square matrix of same order n,

• Property 1:  (A^-1)^-1 = A
• Property 2: (AB)^-1=B^-1A^-1
• Property 3: (A’)^-1= (A^-1)’
• Property 4: (A^m)^-1= (A^-1)^m
• Property 5: |A^-1|= |A|^-1

Example:

=-28+30+18

=20

Similarly we can also obtain the values of B^-1 and A^-1 Then by multiplying B^-1 and A^-1

Applications of Determinants and Matrices:

Consistency of System of Equations

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

Suppose the system of equations is given by:

a₁ x + b₁ y + c₁ z = d₁

a₂ x + b₂ y + c₂ z = d₂

a₃ x + b₃ y + c₃ z = d₃

Now let us say, A, B and X are three matrices, such that;

or  I X = A^–1 B

or  X = A^–1 B

If A is a non-singular matrix, then X = A^-1B.

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

Case-2 :

If A is a singular matrix, then determinant of A, |A| = 0.

Now for such a condition, there exist two cases based on (adj A) B.

• If (adj A) B  O, (O being is zero matrix), then the system of equations does not have a solution and hence is called inconsistent.
• If (adj A) B = O, then the system of equations will have either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

 

Problem :

 Find if the given system of equations is consistent or inconsistent.

x+3y = 5 and 2x + 6y = 8

Solution: Given, the system of equations are:

x+3y = 5 and 2x + 6y = 8

As per the matrix Method, we know;

AX = B

Thus, (adj.A)B ≠ 0

Hence, the given system of equations is inconsistent.