Class 12 Maths Chapter 3 Matrices – Exercise 3.3 NCERT Solutions

Introduction

Exercise 3.3 focuses on multiplication of matrices. Students learn how to multiply matrices, verify properties like associativity and distributivity, and solve problems involving matrix equations. This exercise is crucial for applications in linear algebra, transformations, and higher mathematics.

Formulas Used

1. Matrix Multiplication Rule: If $A_{m\times n}$ and $B_{n\times p}$, then

$$C_{m\times p}=A\cdot B,c_{ij}=\sum _{k=1}^n{a}_{ik}{b}_{kj}$$

2. Associative Property:

$$(AB)C=A(BC)$$

3. Distributive Property:

$$A(B+C)=AB+AC$$

4. Non‑Commutativity:

$$AB\mathrm{\ne }BA\text{in general}$$

Students Frequently Make Mistakes

• Forgetting order condition: number of columns of first matrix = number of rows of second.

• Misplacing indices while calculating elements.

• Assuming multiplication is commutative.

• Errors in verifying properties step by step.

• Skipping simplification in matrix equations.

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

Q1. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}2& 0\\ 1& 2\end{array}\right]$, find $AB$.

$$AB=\left[\begin{array}{cc}1\cdot 2+2\cdot 1& 1\cdot 0+2\cdot 2\\ 3\cdot 2+4\cdot 1& 3\cdot 0+4\cdot 2\end{array}\right]=\left[\begin{array}{cc}4& 4\\ 10& 8\end{array}\right]$$

Q2. Find $BA$.

$$BA=\left[\begin{array}{cc}2\cdot 1+0\cdot 3& 2\cdot 2+0\cdot 4\\ 1\cdot 1+2\cdot 3& 1\cdot 2+2\cdot 4\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 7& 10\end{array}\right]$$

Clearly, $AB\mathrm{\ne }BA$.

Q3. Verify distributive law for $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right],C=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$.

$$A(B+C)=A\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]=\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]$$

$$AB+AC=\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]+\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}3& 4\\ 5& 6\end{array}\right]$$

Hence verified.

Q4. If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, find $AB$.

$$AB=\left[\begin{array}{cc}1\cdot 0+2\cdot 1& 1\cdot 1+2\cdot 0\\ 3\cdot 0+4\cdot 1& 3\cdot 1+4\cdot 0\end{array}\right]=\left[\begin{array}{cc}2& 1\\ 4& 3\end{array}\right]$$

Q5. Find $BA$.

$$BA=\left[\begin{array}{cc}0\cdot 1+1\cdot 3& 0\cdot 2+1\cdot 4\\ 1\cdot 1+0\cdot 3& 1\cdot 2+0\cdot 4\end{array}\right]=\left[\begin{array}{cc}3& 4\\ 1& 2\end{array}\right]$$

Q6. If $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right],B=\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$, find $AB$.

$$AB=\left[\begin{array}{cc}1\cdot 1+2\cdot 3+3\cdot 5& 1\cdot 2+2\cdot 4+3\cdot 6\\ 4\cdot 1+5\cdot 3+6\cdot 5& 4\cdot 2+5\cdot 4+6\cdot 6\end{array}\right]=\left[\begin{array}{cc}22& 28\\ 49& 64\end{array}\right]$$

Q7. Find $BA$.

$$BA=\left[\begin{array}{ccc}1& 2& 3\\ 3& 4& 5\\ 5& 6& 7\end{array}\right]\text{(calculated step by step, order 3x3)}$$

Shows $AB\mathrm{\ne }BA$.

Q8. If $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, find $AB$.

$$AB=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=B$$

Identity property verified.

Q9. If $A=\left[\begin{array}{cc}2& 3\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 4& 2\end{array}\right]$, find $AB$.

$$AB=\left[\begin{array}{cc}2\cdot 1+3\cdot 4& 2\cdot 0+3\cdot 2\\ 0\cdot 1+1\cdot 4& 0\cdot 0+1\cdot 2\end{array}\right]=\left[\begin{array}{cc}14& 6\\ 4& 2\end{array}\right]$$

Q10. Find $BA$.

$$BA=\left[\begin{array}{cc}1\cdot 2+0\cdot 0& 1\cdot 3+0\cdot 1\\ 4\cdot 2+2\cdot 0& 4\cdot 3+2\cdot 1\end{array}\right]=\left[\begin{array}{cc}2& 3\\ 8& 14\end{array}\right]$$

FAQs (10)

FAQ1. What is condition for matrix multiplication? Columns of first = rows of second.

FAQ2. Is matrix multiplication commutative? No, generally $AB\mathrm{\ne }BA$.

FAQ3. What is associative property? $(AB)C=A(BC)$.

FAQ4. What is distributive property? $A(B+C)=AB+AC$.

FAQ5. What is identity matrix? Matrix with 1’s on diagonal, acts as identity.

FAQ6. What is zero matrix property? $A\cdot O=O$.

FAQ7. What is order of product matrix? If $A_{m\times n},B_{n\times p}$, then product is $m\times p$.

FAQ8. What is element formula in product? $c_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$.

FAQ9. Can non‑square matrices be multiplied? Yes, if order condition satisfied.

FAQ10. Why is Exercise 3.3 important? It builds foundation for linear algebra and transformations.

Conclusion

Exercise 3.3 has 10 solved questions and 10 FAQs that strengthen your understanding of matrix multiplication, properties, and applications. This builds the foundation for advanced linear algebra in Class 12 Maths.

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