Class 12 Maths Chapter 3 Matrices – Exercise 3.4 NCERT Solutions

Introduction

Exercise 3.4 focuses on the properties of transpose of matrices. Students learn how to prove identities involving transpose, verify symmetry, and solve problems using transpose rules. This exercise is essential for understanding determinants, orthogonal matrices, and advanced linear algebra.

Formulas Used

1. Transpose Definition:

$$(A^T{)}_{ij}=A_{ji}$$

2. Properties:

   • $(A^T{)}^T=A$

   • $(A+B{)}^T=A^T+B^T$

   • $(kA{)}^T=k{A}^T$

   • $(AB{)}^T=B^T{A}^T$

3. Symmetric Matrix:

$$A^T=A$$

4. Skew‑Symmetric Matrix:

$$A^T=-A$$

Students Frequently Make Mistakes

• Forgetting that transpose reverses multiplication order: $(AB{)}^T=B^T{A}^T$.

• Confusing symmetric with skew‑symmetric matrices.

• Errors in proving properties step by step.

• Misplacing indices while transposing.

• Assuming transpose preserves multiplication order.

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

Q1. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find $A^T$.

$$A^T=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$$

Q2. Verify $(A^T{)}^T=A$ for above matrix.

$$(A^T{)}^T=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=A$$

Q3. If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$, verify $(A+B{)}^T=A^T+B^T$.

$$A+B=\left[\begin{array}{cc}3& 3\\ 4& 6\end{array}\right],(A+B{)}^T=\left[\begin{array}{cc}3& 4\\ 3& 6\end{array}\right]$$

$$A^T+B^T=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\left[\begin{array}{cc}2& 4\\ 3& 5\end{array}\right]=\left[\begin{array}{cc}3& 4\\ 3& 6\end{array}\right]$$

Verified.

Q4. Verify $(kA{)}^T=k{A}^T$ for $k=2,A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$.

$$(2A{)}^T={\left[\begin{array}{cc}2& 4\\ 6& 8\end{array}\right]}^T=\left[\begin{array}{cc}2& 6\\ 4& 8\end{array}\right]$$

$$2A^T=2\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}2& 6\\ 4& 8\end{array}\right]$$

Verified.

Q5. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, verify $(AB{)}^T=B^T{A}^T$.

$$AB=\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right],(AB{)}^T=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$$

$$B^T{A}^T=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$$

Verified.

Q6. Show that $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$ is symmetric.

$$A^T=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]=A$$

Q7. Show that $A=\left[\begin{array}{cc}0& 2\\ -2& 0\end{array}\right]$ is skew‑symmetric.

$$A^T=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]=-A$$

Q8. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$, find $A^T$.

$$A^T=\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$$

Q9. If $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$, check if symmetric.

$$A^T=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=A$$

Yes, symmetric.

Q10. If $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, check if skew‑symmetric.

$$A^T=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=-A$$

Yes, skew‑symmetric.

FAQs (10)

FAQ1. What is transpose of matrix? Interchanging rows and columns.

FAQ2. What is property of double transpose? $(A^T{)}^T=A$.

FAQ3. What is property of transpose of sum? $(A+B{)}^T=A^T+B^T$.

FAQ4. What is property of transpose of scalar multiple? $(kA{)}^T=k{A}^T$.

FAQ5. What is property of transpose of product? $(AB{)}^T=B^T{A}^T$.

FAQ6. What is symmetric matrix? Matrix equal to its transpose.

FAQ7. What is skew‑symmetric matrix? Matrix equal to negative of its transpose.

FAQ8. Can a matrix be both symmetric and skew‑symmetric? Yes, only if all elements are zero.

FAQ9. Why transpose important? Used in determinants, orthogonal matrices, and linear algebra.

FAQ10. Why is Exercise 3.4 important? It builds foundation for properties of matrices and advanced algebra.

Conclusion

Exercise 3.4 has 10 solved questions and 10 FAQs that strengthen your understanding of transpose properties, symmetric and skew‑symmetric matrices. This builds the foundation for advanced linear algebra in Class 12 Maths.

visit:www.fuzymathacademy.com