Class 12 Maths Chapter 3 Matrices – Exercise 3.2 NCERT Solutions

Introduction

Exercise 3.2 focuses on basic operations on matrices such as addition, subtraction, scalar multiplication, and verifying properties of matrix operations. This exercise builds the foundation for matrix algebra and prepares students for advanced applications in determinants and linear equations.

Key Concepts

1. Addition of Matrices: If $A=[a_{ij}]$ and $B=[b_{ij}]$ of same order,

$$A+B=[a_{ij}+b_{ij}]$$

2. Subtraction of Matrices:

$$A-B=[a_{ij}-b_{ij}]$$

3. Scalar Multiplication: For scalar $k$,

$$kA=[k{a}_{ij}]$$

4. Properties of Matrix Addition:

   • Commutative: $A+B=B+A$

   • Associative: $(A+B)+C=A+(B+C)$

   • Existence of zero matrix: $A+O=A$

Students Frequently Make Mistakes

• Trying to add matrices of different orders.

• Forgetting to apply scalar multiplication to every element.

• Confusing subtraction with negative scalar multiplication.

• Misplacing elements during addition/subtraction.

• Ignoring properties of matrix operations.

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

Q1. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$, find $A+B$.

$$A+B=\left[\begin{array}{cc}1+5& 2+6\\ 3+7& 4+8\end{array}\right]=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$$

Q2. If $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],B=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, find $A-B$.

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

Q3. If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, find $3A$.

$$3A=\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$$

Q4. If $A=\left[\begin{array}{cc}2& 4\\ 6& 8\end{array}\right]$, find $-A$.

$$-A=\left[\begin{array}{cc}-2& -4\\ -6& -8\end{array}\right]$$

Q5. Verify commutative property for $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$.

$$A+B=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right],B+A=\left[\begin{array}{cc}6& 8\\ 10& 12\end{array}\right]$$

Hence, $A+B=B+A$.

Q6. Verify associative property for $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right],C=\left[\begin{array}{cc}3& 3\\ 3& 3\end{array}\right]$.

$$(A+B)+C=\left[\begin{array}{cc}6& 5\\ 5& 6\end{array}\right],A+(B+C)=\left[\begin{array}{cc}6& 5\\ 5& 6\end{array}\right]$$

Hence, associative property holds.

Q7. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, find $A+O$.

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

Q8. If $A=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]$, find $2A-3A$.

$$2A-3A=(-A)=\left[\begin{array}{cc}-2& -3\\ -4& -5\end{array}\right]$$

Q9. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}4& 3\\ 2& 1\end{array}\right]$, find $A+B$.

$$A+B=\left[\begin{array}{cc}5& 5\\ 5& 5\end{array}\right]$$

Q10. If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, find $5A$.

$$5A=\left[\begin{array}{cc}5& 5\\ 5& 5\end{array}\right]$$

FAQs (10)

FAQ1. What is matrix addition? Element‑wise addition of matrices of same order.

FAQ2. What is matrix subtraction? Element‑wise subtraction of matrices of same order.

FAQ3. What is scalar multiplication? Multiplying each element of matrix by scalar.

FAQ4. Can we add matrices of different orders? No.

FAQ5. What is zero matrix? Matrix with all elements zero.

FAQ6. What is commutative property of addition? $A+B=B+A$.

FAQ7. What is associative property of addition? $(A+B)+C=A+(B+C)$.

FAQ8. What is additive inverse of matrix? $-A$.

FAQ9. What is identity element in matrix addition? Zero matrix.

FAQ10. Why is Exercise 3.2 important? It builds foundation for matrix algebra operations.

Conclusion

Exercise 3.2 has 10 solved questions and 10 FAQs that strengthen your understanding of matrix operations. This is the next step in mastering matrices in Class 12 Maths.

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