Class 12 Maths Chapter 4 Determinants – Exercise 4.1 NCERT Solutions

Introduction

Exercise 4.1 introduces the concept of determinants of square matrices. Students learn how to evaluate determinants of order 2 and 3, understand properties, and apply them in solving problems. Determinants are fundamental in linear algebra, especially for solving systems of equations and checking invertibility of matrices.

Formulas Used

1. Determinant of 2×2 Matrix:

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

2. Determinant of 3×3 Matrix (Expansion):

$$\mid \begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\mid =a_{11}(a_{22}{a}_{33}-a_{23}{a}_{32})-a_{12}(a_{21}{a}_{33}-a_{23}{a}_{31})+a_{13}(a_{21}{a}_{32}-a_{22}{a}_{31})$$

3. Property: Determinant changes sign if two rows/columns are interchanged.

Students Frequently Make Mistakes

• Forgetting sign convention in expansion.

• Misplacing elements while expanding along row/column.

• Assuming determinant is sum of diagonal elements (wrong).

• Errors in handling negative signs.

• Confusing determinant with matrix itself.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Evaluate $\mid \begin{array}{cc}1& 2\\ 3& 4\end{array}\mid$.

$$=1\cdot 4-2\cdot 3=4-6=-2$$

Q2. Evaluate $\mid \begin{array}{cc}2& 5\\ 7& 3\end{array}\mid$.

$$=2\cdot 3-5\cdot 7=6-35=-29$$

Q3. Evaluate $\mid \begin{array}{cc}1& 0\\ 0& 1\end{array}\mid$.

$$=1\cdot 1-0\cdot 0=1$$

Q4. Evaluate $\mid \begin{array}{cc}3& 2\\ 4& 1\end{array}\mid$.

$$=3\cdot 1-2\cdot 4=3-8=-5$$

Q5. Evaluate $\mid \begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\mid$.

$$=1(5\cdot 9-6\cdot 8)-2(4\cdot 9-6\cdot 7)+3(4\cdot 8-5\cdot 7)$$

$$=1(45-48)-2(36-42)+3(32-35)=-3+12-9=0$$

Q6. Evaluate $\mid \begin{array}{ccc}2& 0& 1\\ 3& 4& 5\\ 1& 2& 3\end{array}\mid$.

$$=2(4\cdot 3-5\cdot 2)-0(3\cdot 3-5\cdot 1)+1(3\cdot 2-4\cdot 1)$$

$$=2(12-10)+0+1(6-4)=2(2)+2=6$$

Q7. Evaluate $\mid \begin{array}{ccc}1& 2& 1\\ 0& 1& 0\\ 2& 3& 4\end{array}\mid$.

$$=1(1\cdot 4-0\cdot 3)-2(0\cdot 4-0\cdot 2)+1(0\cdot 3-1\cdot 2)$$

$$=1(4)-2(0)+1(-2)=4-0-2=2$$

Q8. Evaluate $\mid \begin{array}{ccc}1& -1& 2\\ 2& 3& 1\\ 3& 4& 5\end{array}\mid$.

$$=1(3\cdot 5-1\cdot 4)-(-1)(2\cdot 5-1\cdot 3)+2(2\cdot 4-3\cdot 3)$$

$$=1(15-4)+1(10-3)+2(8-9)=11+7-2=16$$

Q9. Evaluate $\mid \begin{array}{ccc}2& 3& 1\\ 1& 0& 4\\ 3& 2& 5\end{array}\mid$.

$$=2(0\cdot 5-4\cdot 2)-3(1\cdot 5-4\cdot 3)+1(1\cdot 2-0\cdot 3)$$

$$=2(0-8)-3(5-12)+1(2-0)=-16-3(-7)+2=-16+21+2=7$$

Q10. Evaluate $\mid \begin{array}{ccc}4& 2& 1\\ 0& 3& 2\\ 1& 0& 5\end{array}\mid$.

$$=4(3\cdot 5-2\cdot 0)-2(0\cdot 5-2\cdot 1)+1(0\cdot 0-3\cdot 1)$$

$$=4(15-0)-2(0-2)+1(0-3)=60-(-4)-3=61$$

FAQs (10)

FAQ1. What is determinant? A scalar value associated with square matrix.

FAQ2. How to evaluate 2×2 determinant? $ad-bc$.

FAQ3. How to evaluate 3×3 determinant? Expansion along row/column.

FAQ4. What is property of determinant under row interchange? Sign changes.

FAQ5. What is determinant of identity matrix? Always 1.

FAQ6. What is determinant of singular matrix? Zero.

FAQ7. Can determinant be negative? Yes.

FAQ8. What is determinant of triangular matrix? Product of diagonal elements.

FAQ9. Why determinant important? Used in solving equations, invertibility, geometry.

FAQ10. Why is Exercise 4.1 important? It builds foundation for determinants and applications in linear algebra.

Conclusion

Exercise 4.1 has 10 solved questions and 10 FAQs that strengthen your understanding of determinants of 2×2 and 3×3 matrices. This builds the foundation for advanced linear algebra in Class 12 Maths.

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