Determinants, the 4th chapter in the NCERT Solutions for Class 12 Maths, introduces the concept of matrices and the basic laws of matrix algebra. This has innumerable applications in several fields of study.
Topics covered:
1. Introduction
2. Determinant
3. Properties of determinants
4. Area of a triangle
5. Minors and cofactors
6. Adjoint and inverse of a matrix
7. Applications of determinants and matrices

Determinants Class 12 Ncert Solutions elaborates on the concept of matrices and the basic laws adopted while solving algebraic equations. Determinants by definition are unique real number associations of a square matrix.

For example, consider the following matrix A

The determinant of matrix A is as follows IAI:

Let A = [a] be a square matrix of order 1, then det A = I a I and the value of the determinant is the number of itself i.e., I a I = a.

**Example**: Consider the following linear equations:

- A = [5], then det A = I 5 I = 5
- A = [17], then det A = I -7 I = -7

Determinant of a matrix of order two:

For a matrix A of the second order, det A will be equal to a11 I a22 I - a12 I a21 I = a11 a22 - a12 a21

The following are the properties of Determinants Class 12 NCERT Solutions:

- Property 1: If each element in a row or column of a determinant is zero, then the value of the determinant will be zero too. For example, A = [ 1 2 0 ], det A = 0
- Property 2: If each element on one side of the principal diagonal of a determinant is zero, then the value of the determinant is the product of the diagonal elements.
- Property 3: The value of a determinant remains unchanged if its rows and columns are interchanged.
- Property 4: If any two rows or columns of a determinant are interchanged, then the value of the determinant changes by minus sign only.
- Property 5: If two parallel lines (rows or columns) of a determinant are identical (all corresponding elements are the same), then the value of the determinant is zero.
- Property 6: If each element of a determinant’s row or column is multiplied by the same number ‘k’, then the value of the new determinant is ‘k’ times the value of the original determinant.
- Property 7: If each element of a row or column of a determinant consists of a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants whose other rows or columns are not altered.
- Property 8: If we add to each element of a determinant’s row or column the equi-multiples of the corresponding elements of one or more rows or columns, the values of the determinant remains unchanged.
- Property 9: The sum of the product of elements of any row or column with the cofactors of the corresponding elements of some other row or column is zero.
- Property 10: If A and B are square matrices of the same order, then det AB = det A . det B.

It is known that the area of a triangle whose vertices are A(a1, b1), B(a2, b2), and C(c1, c2) is equal to the absolute value of ½ (a1b2 - a2b1 + a2b3 - a3b2 + a3b1 - a1b3).

Hence, the determinant of the above = a1(b2 - b3) - b1 (a2 - a3) + 1(a2b3 - a3b2)

= a1b2 - a2b1 + a2b3 - a3b2 + a3b1 - a1y3.

Thus, the area of triangle ABC is half times the determinant of the above values.

Consider a determinant of the order ‘n’, n>/= 2, then the determinant of order ‘n - 1’ obtained from the determinant after deleting the ‘ith’ row and the ‘jth’ column is called the minor of the lament aij and is usually denoted by Mij where i = 1, 2, 3, …., n and j = 1, 2, 3, …., n.

If Mij is the minor of the element aij in the determinant, then the number (-1)i + j Mij is called the cofactor of the element aij, and is usually denoted by Aij.

Thus, Aij = (-1)1 + j M1j

The sum of the products of the elements of any row or column of a determinant with their corresponding cofactors is equal to the value of the determinant.

- Consider a square matrix A of the order ‘n’, then A is called invertible if and only if there exists a square matrix B of order ‘n’ such that AB = In = BA. Matrix B is called the inverse of A.
- Consider A = [aij] which is a square matrix of order >/ = 2, then the adjoint of A is the transpose of the matrix [Aij] where Aij is the cofactor of the element aij in I A I. It is denoted by adj A.
- The adjoint of a matrix formula is the transpose of the matrix [Aij] where Aij is the cofactor of the element aij in I A I.

For example,

The solution to a system of linear equations:

- This can be done using the inverse of a matrix.

For example, consider the following linear equations:

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

Now, the following determinant can be used to represent the above equations.

A is:

X is:

B is:

- From the above three matrices, the linear equation can be solved using the formula:

X = A-1 B (AX = B → A-1(AX) = A-1 B → InX = A-1B)

- While solving such questions, it is important to know the criteria for consistency or inconsistency:
- If I A I is not equal to zero, then the system is consistent and it has a unique solution given by AX = B.
- If I A I is equal to zero and (adj A) B is equal to zero, the system is consistent and has infinitely many solutions.
- If I A I is equal to zero and (adj A) B is not equal to zero, the system is inconsistent.

__Example 1__: Evaluate the following.

__Answer__: Here, the 2nd row can be extrapolated to multiply with the individual determinants as follows:

5 (-14-143) + 0 - 0 = -785.

__Example 2__: Given that A is a 3 x 3 matrix with I 3A I = k I A I, find the value of the constant ‘k’?

__Answer__: Here, we know that if A is a square matrix of order ‘n’, then I kA I = kn I A I.

Therefore, I 3A I = 33 I A I = 27 I A I.

I 3A I = k I A I

Hence, k = 27

Determinants Class 12 Ncert Solutions is an essential subject as well as a difficult one with many problems, diagrams and concepts.

Key points of this chapter are:

- Easy language and eye-catching formats.
- Topics as per the latest syllabus pattern so students can revise the notes in minimum time with maximum accuracy.
- Follows the guidelines of the CBSE syllabus.
- After studying Determinants Class 12 NCERT Solutions notes, students will be more confident when presented with other enormous textbooks.
- Covers all necessary formulas and concepts presented in the chapter.
- On studying these notes, students won’t have to spend much time revising before the exams.

**1. How many problems are there in NCERT Solutions for Class 12 Maths Chapter 4? **

There are 8 sums in the first exercise, Ex.-4.1, and 16 sums in the second exercise, Ex.-4.2. There are 5 sums in the third exercise, Ex.-4.3, 5 sums in the fourth exercise, Ex.-4.4., 18 sums in the fifth exercise Ex.-4.5, and 16 sums in the last exercise, Ex.-4.6. The last exercise is followed by a miscellaneous exercise, and there are 18 sums in the miscellaneous exercise for this chapter. All the sums are solved and explained in the NCERT Solutions for Class 12 Maths Chapter 4.

**2. What do you understand by the term determinants? **

Determinants by definition are unique real number associations of a square matrix. They can be used to solve linear equations. Using the properties of adjoints, several linear equations derived from matrices of unequal orders can be operated on.

**3. Can you suggest video content for determinants? **

The videos related to class 12 NCERT Maths can be accessed from this website whenever necessary. They are valuable learning assets that can solve any query. In a nutshell, they can improve performance and become part of an interactive learning session. They speed up a student’s problem-solving skills and clarify doubts by simplifying difficult concepts.

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