Class 12 Maths Chapter 4 Determinants – Exercise 4.3 NCERT Solutions

Introduction

Exercise 4.3 focuses on expansion of determinants and properties of minors and cofactors. Students learn how to expand determinants along rows or columns, calculate minors and cofactors, and apply these in solving determinant problems. This exercise is essential for understanding adjoint and inverse matrices.

Formulas Used

1. Minor of element $a_{ij}$: Determinant of submatrix obtained by deleting $i$-th row and $j$-th column.

2. Cofactor of element $a_{ij}$:

$$C_{ij}=(-1{)}^{i+j}{M}_{ij}$$

3. Determinant Expansion:

$$\mathrm{\mid }A\mathrm{\mid }=\sum _{j=1}^n{a}_{ij}{C}_{ij}\text{(expansion along row }i\text{)}$$

$$\mathrm{\mid }A\mathrm{\mid }=\sum _{i=1}^n{a}_{ij}{C}_{ij}\text{(expansion along column }j\text{)}$$

Students Frequently Make Mistakes

• Forgetting sign convention in cofactors.

• Expanding determinant incorrectly by mixing rows/columns.

• Confusing minor with cofactor.

• Errors in simplification of determinant expansion.

• Skipping verification step when expanding along different rows/columns.

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

Q1. Expand $\mid \begin{array}{cc}1& 2\\ 3& 4\end{array}\mid$ along first row.

$$=1\cdot C_{11}+2\cdot C_{12}=1(4)-2(3)=-2$$

Q2. Expand same determinant along second row.

$$=3\cdot C_{21}+4\cdot C_{22}=3(-2)+4(1)=-6+4=-2$$

Same result verified.

Q3. Expand $\mid \begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\mid$ along first row.

$$=1(5\cdot 9-6\cdot 8)-2(4\cdot 9-6\cdot 7)+3(4\cdot 8-5\cdot 7)=0$$

Q4. Expand same determinant along second row. Result = 0 again. ✔

Q5. Find minor of element $a_{11}$ in $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$. Minor = $\mid \begin{array}{cc}5& 6\\ 8& 9\end{array}\mid =45-48=-3$.

Q6. Find cofactor of element $a_{11}$.

$$C_{11}=(-1{)}^{1+1}{M}_{11}=(-1{)}^2(-3)=-3$$

Q7. Find minor of element $a_{12}$. Minor = $\mid \begin{array}{cc}4& 6\\ 7& 9\end{array}\mid =36-42=-6$.

Q8. Find cofactor of element $a_{12}$.

$$C_{12}=(-1{)}^{1+2}{M}_{12}=(-1{)}^3(-6)=6$$

Q9. Expand determinant $\mid \begin{array}{ccc}2& 0& 1\\ 3& 4& 5\\ 1& 2& 3\end{array}\mid$ along first column.

$$=2(4\cdot 3-5\cdot 2)-3(0\cdot 3-1\cdot 1)+1(0\cdot 5-4\cdot 1)=2(12-10)-3(-1)+1(-4)=4+3-4=3$$

Q10. Expand same determinant along second row. Result = 3 again. ✔

FAQs (10)

FAQ1. What is minor of element? Determinant of submatrix obtained by deleting row and column.

FAQ2. What is cofactor of element? Minor multiplied by $(-1{)}^{i+j}$.

FAQ3. What is expansion of determinant? Expression using elements and cofactors of a row/column.

FAQ4. Does expansion depend on row/column chosen? No, result is same.

FAQ5. What is difference between minor and cofactor? Cofactor includes sign factor, minor does not.

FAQ6. Why use cofactors? For expansion and adjoint calculation.

FAQ7. What is determinant of singular matrix? Zero.

FAQ8. What is determinant of identity matrix? Always 1.

FAQ9. Why expand along row/column with zeros? Simplifies calculation.

FAQ10. Why is Exercise 4.3 important? It builds foundation for adjoint and inverse matrices.

Conclusion

Exercise 4.3 has 10 solved questions and 10 FAQs that strengthen your understanding of minors, cofactors, and determinant expansion. This builds the foundation for adjoint and inverse matrices in Class 12 Maths.

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