Class 12 Chapter 4 (determinants) class notes

Chapter - 4

  Determinants 

☞  System of algebraic equations can be expressed in the form of matrices. 

         • Linear Equations Format 

             a1x+b1y=c1

             a2x+b2y=c2

        • Matrix Format:

☞ The values of the variables satisfying all the linear equations in the system, is called solution of system of linear equations. 

☞ If the system of linear equations has a unique solution. This unique solution is called determinant of Solution or det A

☞ Applications of Determinants 

☞ Engineering 

☞ Science 

☞ Economics  

☞ Social Science, etc.

☸ Determinant 

♦ A determinant is defined as a (mapping) unction from the set o square matrices to the set of real numbers 

♦ Every square matrix A is associated with a number, called its determinant 

♦ Denoted by det (A) or |A| or ∆

♦ Only square matrices have determinants.

♦ The matrices which are not square do not have determinants 

♦ For matrix A, |A| is read as determinant of A and not modulus of A. 

☸ Types of Determinant  

♦ 1. First Order Determinant  

  ♦ Let A = [a ] be the matrix of order 1, then determinant of A is defined to be equal to a 

  ♦ If A = [a], then det (A) = |A| = a

♦ 2. Second Order Determinant 

♦ 3. Third Order Determinant 

♦ Can be determined by expressing it in terms of second order determinants

The below method is explained for expansion around Row 1 

The value of the determinant, thus will be the sum of the product of element in line parallel to the diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus,

The same procedure can be repeated for Row 2, Row 3, Column 1, Column 2, and Column 3 

♦ Note  

   ♦ Expanding a determinant along any row or column gives same value.

   ♦ This method doesn't work for determinants of order greater than 3. 

   ♦ For easier calculations, we shall expand the determinant along that row or column which contains maximum number of zeros 

   ♦ In general, if A = kB where A and B are square matrices of order n, then | A| = kⁿ |B |, where n = 1, 2, 3

☸ Properties of Determinants 

♦ Helps in simplifying its evaluation by obtaining maximum number of zeros in a row or a column. 

♦ These properties are true for determinants of any order. 

☸ Property 1 

♦ The value of the determinant remains unchanged if its rows and columns are interchanged 

♦ Verification:

Expanding ∆₁ along first column, we get 

∆₁ =a₁ (b₂ c₃ - c₂ b₃) - a₂(b₁ c₃ - b₃ c₁) + a₃ (b₁ c₂ - b₂ c₁)

Hence ∆ = ∆₁

♦ Note:

♦ It follows from above property that if A is a square matrix, 

     Then det (A) = det (A'), where A' = transpose of A 

♦ If Ri = ith row and Ci = ith column, then for interchange of row and 

♦ columns, we will symbolically write Ci⇔Ri

☸ Property 2 

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

♦ Verification :

☸ Property 3 

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

♦ Verification:

♦ If we interchange the identical rows (or columns) of the determinant ∆, then ∆ does not change.

♦ However, by Property 2, it follows that ∆ has changed its sign 

♦ Therefore ∆ = -∆ or ∆ = 0

☸  Property 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

♦ Verification

☸ Property 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.

♦ Verification:

☸ Property 6

♦ If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation

☸ Property 7 

♦ If each element of a row (or column) of a determinant is zero, then its value is zero

☸ Property 8 

♦ In a determinant, If all the elements on one side of the principal diagonal are Zero's , then the value of the determinant is equal to the product of the elements in the principal diagonal

☸ Area of a Triangle 

♦ Let (x₁,y₁), (X₂, y₂), and (x₃, y₃) be the vertices of a triangle, then

☸ Note

♦ Area is a positive quantity, we always take the absolute value of the determinant .
♦ If area is given, use both positive and negative values of the determinant for caleulation.
♦ The area of the triangle formed by three collinear points is zero.

☸ Minors and Cofactors 

☸ Minor

♦ If the row and column containing the element a₁₁ (i.e., 1st row and 1st column)are removed, we get the second order determinant which is called the Minor of element a₁₁
♦ Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column which element aij lies.

♦ Minor of an element aij is denoted by Mij
♦ Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order n-1

♦ Eg: Find Minor o the element 6 in the determinant A given

☸ Cofactor 

♦ If the minors are multiplied by the proper signs we get cofactors
♦The cofactor of the element aij is Cij = (-1) Mij
♦The signs to be multiplied are given by the rule

♦ Cofactor of 4 is A₁₂ =(-1) M₁₂ =(-1)³(4) =-4

☸ Adjoint and Inverse of a Matrix 

♦ Adjoint of matrix is the transpose of the matrix of cofactors of the given matrix

☸ Theorem 1 

♦ If A be any given square matrix of order n, 

 Then A (adj A) = (adj A) A = A I,

 Where I is the identity matrix of order n 

♦ Verification:

Similarly, we can show (adj A) A = AI
Hence A (adj A) = (adj A) A = AI

☸ Singular & No Singular Matrix:

♦ A square matrix A is said to be singular if |A| = o
♦ A square matrix A is said to be non-singular if |A |≠ 0

☸ Theorem 2 

♦ If A and B are non-singular matrices of the same order, then AB and BA are also non- singular matrices of the same order.

☸ Theorem 3 

♦ 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 4 

&dsiams; A square matrix A is invertible if and only if A is non-singular matrix.

♦ Verification

 Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that AB = BA = I

Now AB = I. So |AB| = I or |A| |B| = 1 (since |I|= 1, |AB|=| A||B|). This gives |A| ≠ 0. Hence A is non-singular.

Conversely, let A be non-singular. Then |A| ≠ 0
Now A (adj A) = (adj A) A = |A| I (Theorem 1)

☸ Applications of Determinants and Matrices 

♦ Used for solving the system of linear equations in two or three variables and for checking the consistency of the system of linear 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 

♦ Let the system of Equations be as below:

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

☸ Case I

If A is a non-singular matrix, then its inverse exists.

AX = B
A⁻¹(AX) = A⁻¹B (premultiplying by A⁻¹)
(A⁻¹A)X -A⁻¹B (by associative property)
1X = A⁻¹B
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

𘗈  Case II 

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 solution does not exist and the system of equations is called inconsistent.
If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution

☸  Summary

For a square matrix A in matrix equation AX = B

♦ |A| ≠ 0, there exists unique solution
♦ |A| = 0 and (adj A) B ≠ 0, then there exists no solution
♦ |A| o and (adj A) B = 0, then system may or may not be consistent.



   

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