Class 12 Math Chapter 3 Matrices – Complete NCERT Solutions

Introduction

Matrices are one of the most powerful tools in mathematics. They simplify solving systems of linear equations, represent geometric transformations like rotation and reflection, and are widely used in computer science, economics, genetics, and cryptography. In this chapter, we study the basics of matrices, their types, operations, and properties.

Key Definitions and Formulas

• Matrix: Rectangular array of numbers/functions.

• Order of Matrix: If a matrix has $m$ rows and $n$ columns, it is of order $m\times n$.

• Types of Matrices:

  • Row Matrix: One row.

  • Column Matrix: One column.

  • Square Matrix: Rows = Columns.

  • Diagonal Matrix: Non‑diagonal entries = 0.

  • Scalar Matrix: Diagonal entries equal.

  • Identity Matrix: Diagonal entries = 1, rest = 0.

  • Zero Matrix: All entries = 0.

• Equality of Matrices: Two matrices are equal if they are of same order and corresponding elements are equal.

• Addition: $C=A+B$, element‑wise addition.

• Scalar Multiplication: $kA=[k{a}_{ij}]$.

• Matrix Multiplication: Defined if columns of $A$ = rows of $B$.

$$(AB{)}_{ik}=\sum _{j=1}^n{a}_{ij}{b}_{jk}$$

Solved Examples from NCERT

Example 1: Represent men and women workers in 3 factories as a $3\times 2$ matrix. Solution:

$$A=\left[\begin{array}{cc}30& 25\\ 25& 31\\ 27& 26\end{array}\right]$$

Entry (3,2) = 26 (women workers in factory III).

Example 2: Matrix with 8 elements. Possible orders: $1\times 8,8\times 1,2\times 4,4\times 2$.

Example 6: Add matrices

$$A=\left[\begin{array}{ccc}1& 3& -1\\ 2& 1& 0\end{array}\right],B=\left[\begin{array}{ccc}-2& 5& 1\\ 3& 1& 2\end{array}\right]$$

$$A+B=\left[\begin{array}{ccc}-1& 8& 0\\ 5& 2& 2\end{array}\right]$$

Example 12: Multiply

$$A=\left[\begin{array}{cc}6& 9\\ 2& 3\end{array}\right],B=\left[\begin{array}{ccc}2& 6& 0\\ 7& 9& 8\end{array}\right]$$

$$AB=\left[\begin{array}{ccc}75& 117& 72\\ 25& 39& 24\end{array}\right]$$

Example 13: Show non‑commutativity: $AB\mathrm{\ne }BA$.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 3.1 (2): Matrix with 24 elements. Possible orders: $1\times 24,24\times 1,2\times 12,12\times 2,3\times 8,8\times 3,4\times 6,6\times 4$.

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

1. Q: What is a matrix? A: Rectangular array of numbers/functions.

2. Q: What is order of matrix with 3 rows and 4 columns? A: $3\times 4$.

3. Q: What is a row matrix? A: Matrix with one row.

4. Q: What is a column matrix? A: Matrix with one column.

5. Q: What is a square matrix? A: Rows = Columns.

6. Q: What is a diagonal matrix? A: Non‑diagonal entries = 0.

7. Q: What is a scalar matrix? A: Diagonal entries equal.

8. Q: What is identity matrix? A: Diagonal entries = 1, rest = 0.

9. Q: What is zero matrix? A: All entries = 0.

10. Q: When are two matrices equal? A: Same order, equal corresponding elements.

11. Q: Define matrix addition. A: Element‑wise addition.

12. Q: Define scalar multiplication. A: Multiply each element by scalar.

13. Q: Define matrix multiplication. A: Row of first × column of second.

14. Q: Is matrix multiplication commutative? A: No, generally $AB\mathrm{\ne }BA$.

15. Q: Is matrix multiplication associative? A: Yes, $(AB)C=A(BC)$.

16. Q: What is additive identity in matrices? A: Zero matrix.

17. Q: What is additive inverse in matrices? A: Negative of matrix.

18. Q: Can product of two non‑zero matrices be zero matrix? A: Yes.

19. Q: How many $3\times 3$ matrices with entries 0 or 1? A: $2^9=512$.

20. Q: What is scalar multiple of identity matrix? A: Scalar matrix.

Conclusion

Matrices are fundamental in mathematics and applications across science and engineering. By mastering types, operations, and properties, you can solve NCERT problems and apply matrix algebra in real‑world contexts.

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