Determinants:Class 12 Maths NCERT Chapter 4

Key Features of NCERT Material for Class 12 Maths Chapter 4 – Determinants

In the previous Chapter 3:Matrices we have learned about matrix and its different properties.In this Chapter 4:Determinants we will learn about determinants and properties of determinants.

Quick revision notes

Determinant: Determinant is the mathematical value of the square matrix. In this way, to each square matrix A = [aij] of request n, we can relate a number (genuine or complex) considered determinant of the square matrix A. It is signified by det An or |A|.

Note

(I) Read |A| as determinant A and not absolute value of A.

(ii) Determinant gives mathematical value yet matrix don’t give mathematical value.

(iii) A determinant consistently has an equivalent number of lines and sections, for example just square matrix have determinants.

Value of a Determinant

Value of a determinant of a matrix of order 2, A = is

Value of  a determinant of a matrix of order 3, A =   is given by expressing it in terms of second order determinant. This is called as expansion of a determinant along a row (or column).

Note

(i) For easier calculations of a determinant, we shall expand the determinant along that row or column which contains the maximum number of zeroes.

(ii) While expanding, instead of multiplying by (-1)i+j, we can multiply by +1 or -1 according to as (i + j) is even or odd.

Let A be a matrix of order say n and let |A| = x. Then, |kA| = kn |A| = kn x, where n = 1, 2, 3,…

Minor: Minor of an element ay of a determinant, is a determinant which is obtained by deleting the ith row and jth column in which element ay lies. Minor of an element aij is then denoted by Mij.

Note: Minor of an element of a determinant of order n(n ≥ 2) is a then determinant of order (n – 1).

Cofactor: Cofactor of an element aij of a determinant, denoted by Aij or Cij is defined as Aij = (-1)i+j Mij, where Mij is a minor of an element aij.

Note

(i) For expanding the determinant, we minors and cofactors can be used  as

(ii) If elements of a row (or column) are multiplied with cofactors of another row (or column), then their sum is zero.

Singular and non-singular Matrix: If the value of a determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.

Theorems

(i) 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.

(ii) The determinant of the product of matrices is always equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.

Adjoint of a Matrix: The adjoint of a square matrix ‘A’ is the transpose of the matrix which is obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).

In general, adjoint of a matrix A = [aij]n×n is a matrix [Aji]n×n, where Aji is a cofactor of element aji.

Properties of Adjoint of a Matrix

If A is a square matrix of order say n × n, then

The area of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by

NOTE: Since the area is a positive quantity we always take the absolute value of the determinant.

Properties of Determinants

To discover the value of the determinant, we attempt to make the maximum conceivable zero out of a line (or a section) by utilizing properties given underneath and afterward grow the determinant comparing that line (or segment).

Following are the different properties of determinants:

1. In the event that all the components of any line or section of a determinant are zero, at that point the value of a determinant is zero.
2. On the off chance that every component of any one line or one section of a determinant is a various of scalar k, at that point the value of the determinant is a different of k. at that point the value of the determinant is a various of k. for example

3. On the off chance that in a determinant any two lines or segments are interchanged, at that point the value of the determinant acquired is negative of the value of the given determinant. On the off chance that we make n such changes of lines (sections) indeterminant ∆ and get determinant ∆ , at that point ∆1 = (- 1)n ∆.

4. In the event that all comparing components of any two lines or segments of a determinant are indistinguishable or proportional, at that point the value of the determinant is zero.

[∴ R1 and R3 are identical.]

1. The value of a determinant stays unchanged on changing lines into sections and segments into columns. It follows that, in the event that A will be a square matrix, at that point |A’| = |A|.

Note: det(A) = det(A’), where A’ = transpose of A.

1. Assuming a few or all components of a line or segment of a determinant are communicated as a sum of at least two terms, at that point the determinant can be communicated as the sum of at least two determinants, for example

7. In the components of any line or segment of a determinant, on the off chance that we include or take away the products of comparing components of some other line or section, at that point the value of determinant stays unchanged, for example

At the end of the day, the value of determinants continues as before, on the off chance that we apply the operation Ri → Ri + kEj or Ci → Cj → kCj.

Inverse of a Matrix and Applications of Determinants and Matrix

1. Inverse of a Square Matrix: If A is a non-singular matrix (i.e. |A| ≠ 0), then

Note: Inverse of a matrix, if exists, is unique.

Properties of a Inverse Matrix

• (A-1)-1 = A
• (AT)-1=(A-1)T
• (AB)-1 = B-1A-1
• (ABC)-1 =C-1B-1A-1
1. Solution of system of linear equations using inverse of a matrix.

Let the given system of equations be a1x + b1y + c1z = d1; a2x + b2y + c2z = d2 and a3x + b3y + c3z = d3.

We compose the following system of linear equations in matrix form as AX = B, where

Case I: If |A| ≠ 0, at that point the system is consistent and has a unique solution which is given by X = A-1B.

Case II: If |A| = 0 and (adj A) B ≠ 0, at that point the system is inconsistent and has no solution.

Case III: If |A| = 0 and (adj A) B = 0, at that point system may be either consistent or inconsistent according to as the system have either infinitely many solutions or no solutions.

October 29, 2020