Class 12 Maths Chapter 4 Determinants – Exercise 4.2 NCERT Solutions

Introduction

Exercise 4.2 focuses on the properties of determinants. Students learn how to apply determinant properties such as row/column operations, linearity, and sign changes to simplify calculations. These properties are essential for solving higher‑order determinants quickly and efficiently.

Formulas Used

1. Row/Column Interchange:

$$\text{Interchanging two rows (or columns) changes sign of determinant.}$$

2. Row/Column Multiplication:

$$\text{If a row (or column) is multiplied by }k,\text{ determinant is multiplied by }k\mathrm{.}$$

3. Row/Column Addition:

$$\text{Adding a multiple of one row (or column) to another does not change determinant.}$$

4. Determinant of Triangular Matrix:

$$\text{Product of diagonal elements.}$$

Students Frequently Make Mistakes

• Forgetting sign change when rows/columns are interchanged.

• Multiplying determinant incorrectly when scaling a row.

• Confusing property of addition with multiplication.

• Errors in simplifying determinants using properties.

• Assuming determinant remains unchanged under all operations.

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

Q1. Show that $\mid \begin{array}{cc}1& 2\\ 3& 4\end{array}\mid =-\mid \begin{array}{cc}3& 4\\ 1& 2\end{array}\mid$. Interchanging rows changes sign. ✔

Q2. Evaluate $\mid \begin{array}{cc}2& 3\\ 4& 6\end{array}\mid$. Rows proportional → determinant = 0.

Q3. Evaluate $\mid \begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\mid$. Rows linearly dependent → determinant = 0.

Q4. Show that $\mid \begin{array}{cc}a& b\\ c& d\end{array}\mid =\mid \begin{array}{cc}a& b\\ c+ka& d+kb\end{array}\mid$. Adding multiple of one row to another does not change determinant. ✔

Q5. Evaluate $\mid \begin{array}{ccc}1& 0& 0\\ 2& 3& 0\\ 4& 5& 6\end{array}\mid$. Triangular matrix → product of diagonal = $1\cdot 3\cdot 6=18$.

Q6. Evaluate $\mid \begin{array}{ccc}2& 0& 0\\ 0& 3& 0\\ 0& 0& 4\end{array}\mid$. Diagonal matrix → product of diagonal = $2\cdot 3\cdot 4=24$.

Q7. Show that determinant remains unchanged if we add multiple of one column to another. Property verified. ✔

Q8. Evaluate $\mid \begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\mid$. Columns proportional → determinant = 0.

Q9. Evaluate $\mid \begin{array}{ccc}1& 2& 3\\ 0& 1& 4\\ 0& 0& 2\end{array}\mid$. Upper triangular → product of diagonal = $1\cdot 1\cdot 2=2$.

Q10. Show that $\mid \begin{array}{cc}a& b\\ c& d\end{array}\mid =\mid \begin{array}{cc}ka& kb\\ c& d\end{array}\mid \mathrm{/}k$. Scaling row multiplies determinant by $k$. ✔

FAQs (10)

FAQ1. What happens if two rows are equal? Determinant = 0.

FAQ2. What happens if two rows are proportional? Determinant = 0.

FAQ3. What happens if two rows are interchanged? Sign changes.

FAQ4. What happens if a row is multiplied by $k$? Determinant multiplied by $k$.

FAQ5. What happens if a multiple of one row is added to another? Determinant unchanged.

FAQ6. What is determinant of triangular matrix? Product of diagonal elements.

FAQ7. What is determinant of identity matrix? Always 1.

FAQ8. What is determinant of zero matrix? Always 0.

FAQ9. Why use properties of determinants? To simplify calculations.

FAQ10. Why is Exercise 4.2 important? It builds foundation for solving determinants efficiently in higher algebra.

Conclusion

Exercise 4.2 has 10 solved questions and 10 FAQs that strengthen your understanding of determinant properties. This builds the foundation for advanced applications in Class 12 Maths.

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