Chapter 3: Matrices

A matrix is a rectangular table of numbers. The idea looks simple, but the algebra of matrices , addition, multiplication, transpose, inverse , encodes nearly every linear transformation in science. The chapter introduces matrices as objects of arithmetic, defines their rules of combination, and uses them to package systems of linear equations into a single equation $AX = B$.

The reason matrices matter is that many real-world systems are linear in their unknowns. If you know how to set up and solve $AX = B$, you can handle everything from balancing chemical equations to fitting a regression line. In Class XII the focus is on $2 \times 2$ and $3 \times 3$ matrices and on the mechanics of computing , but the techniques scale to any size.

Two features distinguish matrix algebra from ordinary algebra. First, multiplication is not commutative: $AB \neq BA$ in general. Second, not every nonzero matrix has an inverse; only those with nonzero determinant do. These two facts force you to keep careful track of order and to develop intuition for when an operation is reversible.

Board questions in this chapter are heavy on calculation: add these matrices, multiply, find the inverse using elementary row operations, verify $A(BC) = (AB)C$. JEE questions add structural twists: prove $A$ is invertible iff a certain matrix identity holds, or compute $A^n$ for large $n$ using the Cayley-Hamilton theorem (preview).

A practical tip: always read the size of a matrix first. The product $AB$ exists iff the number of columns of $A$ equals the number of rows of $B$. Many errors begin with a forgotten dimension check.

This chapter feeds directly into determinants (Chapter 4), which give a number summarising the matrix; and into systems of equations, which can be solved using matrix inverses.

What's inside

1. Matrix definition and types , rectangular arrays, special shapes.
2. Operations on matrices , addition, scalar multiplication, multiplication.
3. Transpose and symmetric matrices , and the symmetric/skew decomposition.
4. Elementary operations , row reductions.
5. Invertible matrices , definition and computation by row operations.

Key results / Formula card

Matrix multiplication is associative and distributive over addition, but not commutative.

How to read this chapter

Start with the definitions and types. Practise multiplication until you can do $2 \times 2$ products in your head. Then learn elementary row operations and finding inverses; this skill becomes a staple in higher mathematics.