Class 12 Math Chapter 4 Determinants – Complete NCERT Solutions

Introduction

Determinants are closely linked with matrices and play a vital role in solving systems of linear equations, finding areas of triangles, and checking consistency of equations. They are widely used in engineering, economics, and science. In this chapter, we study determinants up to order three, their properties, minors, cofactors, adjoint, inverse, and applications.

Key Formulas

• Determinant of 2×2 matrix:

$$\mid \begin{array}{cc}a& b\\ c& d\end{array}\mid =ad-bc$$

• Determinant of 3×3 matrix (expansion along first row):

$$\mathrm{\mid }A\mathrm{\mid }=a_{11}(a_{22}{a}_{33}-a_{23}{a}_{32})-a_{12}(a_{21}{a}_{33}-a_{23}{a}_{31})+a_{13}(a_{21}{a}_{32}-a_{22}{a}_{31})$$

• Area of triangle with vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$:

$$\text{Area}=\frac{1}{2}\mid \mid \begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\mid \mid$$

• Minor of element $a_{ij}$: Determinant obtained by deleting ith row and jth column.

• Cofactor: $A_{ij}=(-1{)}^{i+j}{M}_{ij}$.

• Adjoint: Transpose of cofactor matrix.

• Inverse:

$$A^{-1}=\frac{1}{\mathrm{\mid }A\mathrm{\mid }}\text{adj}(A),\text{if }\mathrm{\mid }A\mathrm{\mid }\mathrm{\ne }0$$

Solved Examples from NCERT

Example 1: Evaluate

$$\mid \begin{array}{cc}2& 4\\ -5& -1\end{array}\mid =2(-1)-(-5)(4)=-2+20=18$$

Example 3: Evaluate determinant

$$\mid \begin{array}{ccc}-1& 3& 0\\ 1& 2& 4\\ 4& 1& 0\end{array}\mid$$

Expanding along third column: result = $-52$.

Example 6: Find area of triangle with vertices (3,8), (-4,2), (5,1).

$$\text{Area}=\frac{1}{2}\mid \mid \begin{array}{ccc}3& 8& 1\\ -4& 2& 1\\ 5& 1& 1\end{array}\mid \mid =\frac{1}{2}\mathrm{\mid }-61\mathrm{\mid }=30.5$$

Example 13: Verify $A\cdot \text{adj}(A)=\mathrm{\mid }A\mathrm{\mid }I$ and find inverse.

$$A^{-1}=\frac{1}{\mathrm{\mid }A\mathrm{\mid }}\text{adj}(A)$$

Example 16: Solve system

$$2x+5y=1,3x+2y=7$$

Solution using matrix inverse: $x=3,y=-1$.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 4.1 (1):

$$\mid \begin{array}{cc}2& 4\\ -5& -1\end{array}\mid =18$$

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

1. Q: What is determinant of $\mid \begin{array}{cc}a& b\\ c& d\end{array}\mid$? A: $ad-bc$.

2. Q: Can non‑square matrices have determinants? A: No, only square matrices.

3. Q: What is minor of element $a_{ij}$? A: Determinant after deleting ith row and jth column.

4. Q: What is cofactor of element $a_{ij}$? A: $A_{ij}=(-1{)}^{i+j}{M}_{ij}$.

5. Q: What is adjoint of matrix? A: Transpose of cofactor matrix.

6. Q: When does inverse of matrix exist? A: When determinant ≠ 0.

7. Q: What is singular matrix? A: Matrix with determinant = 0.

8. Q: What is non‑singular matrix? A: Matrix with determinant ≠ 0.

9. Q: Formula for inverse of 2×2 matrix? A:

$$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$$

10. Q: What is property of determinant under scalar multiplication? A: $\mathrm{\mid }kA\mathrm{\mid }=k^n\mathrm{\mid }A\mathrm{\mid }$.

11. Q: What is area of triangle using determinant? A: $\frac{1}{2}\mid \mid \begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\mid \mid$.

12. Q: What is expansion of determinant along row? A: Sum of products of elements with cofactors.

13. Q: What is property of determinant of product? A: $\mathrm{\mid }AB\mathrm{\mid }=\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }$.

14. Q: What is property of adjoint determinant? A: $\mathrm{\mid }\text{adj}(A)\mathrm{\mid }=\mathrm{\mid }A{\mathrm{\mid }}^{n-1}$.

15. Q: What is consistency of system of equations? A: If solution exists.

16. Q: What is inconsistency? A: If solution does not exist.

17. Q: How to solve system using matrix method? A: Write AX=B, then X=A⁻¹B.

18. Q: What is Cayley‑Hamilton theorem application? A: To find inverse using characteristic equation.

19. Q: What is determinant of identity matrix? A: 1.

20. Q: What is determinant of zero matrix? A: 0.

Conclusion

Determinants are a powerful tool in linear algebra. They help in solving equations, finding areas, and checking consistency. By mastering minors, cofactors, adjoint, and inverse, you can tackle NCERT problems and apply these concepts in real‑life contexts.

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