What this chapter is about
Matrices are rectangular arrangements of numbers, symbols or expressions organised in rows and columns. This chapter introduces matrices as mathematical objects that can be added, subtracted, multiplied and transformed according to precise rules. A Class 12 student meets matrices now because they provide a compact way to represent and solve systems of linear equations, perform transformations in geometry, and handle large data sets in economics, physics and computer science.
After studying this chapter, you should be able to recognise different types of matrices, perform arithmetic operations on them, find the transpose of a matrix, and understand when two matrices can be multiplied. You will also learn about symmetric and skew-symmetric matrices, and how every square matrix can be expressed as a sum of these two special types. These skills prepare the ground for determinants and the matrix method of solving equations in later work.
The power of matrices lies in their ability to compress information. Instead of writing nine separate equations, you can write one matrix equation. This economy of notation becomes essential in higher mathematics, engineering and data science.
Key ideas
- A matrix is an ordered rectangular array of numbers arranged in m rows and n columns; its order is written as m × n.
- Two matrices are equal if and only if they have the same order and every corresponding element is equal.
- Addition and subtraction of matrices are defined only for matrices of the same order; you add or subtract corresponding elements.
- Scalar multiplication means multiplying every element of a matrix by the same number (scalar).
- Matrix multiplication A × B is defined only when the number of columns in A equals the number of rows in B; the resulting matrix has order (rows of A) × (columns of B).
- Matrix multiplication is associative and distributive over addition, but it is not commutative: in general, AB ≠ BA.
- The transpose of a matrix A, written Aᵀ or A′, is formed by interchanging its rows and columns.
- A square matrix A is symmetric if A′ = A, and skew-symmetric if A′ = −A; every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.
Formulas and facts to remember
- Order of a matrix with m rows and n columns: m × n; it has m × n elements.
- Element in the iᵗʰ row and jᵗʰ column of matrix A is denoted aᵢⱼ.
- (A + B)′ = A′ + B′ (transpose of a sum equals sum of transposes).
- (kA)′ = kA′ for any scalar k.
- (AB)′ = B′A′ (transpose of a product reverses the order).
- (A′)′ = A (transpose of transpose returns the original matrix).
- For a square matrix A: symmetric part = (1/2)(A + A′); skew-symmetric part = (1/2)(A − A′).
- Identity matrix Iₙ of order n × n has 1 on every diagonal entry and 0 elsewhere; AI = IA = A when multiplication is defined.
- Zero matrix O has every element 0; A + O = A.
Worked examples
Example 1: Finding elements and order
A matrix P has 12 elements. List all possible orders it can have.
Solution
The product of rows and columns must equal 12. Possible factor pairs: 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, 12 × 1. So the possible orders are 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2 and 12 × 1.
Example 2: Matrix multiplication
Let A be a 2 × 3 matrix with rows [1, 2, 3] and [4, 5, 6]. Let B be a 3 × 2 matrix with rows [1, 0], [0, 1], [1, 1]. Find the product AB.
Solution
A is 2 × 3 and B is 3 × 2. Number of columns in A equals number of rows in B (both 3), so AB is defined and has order 2 × 2.
Element (1,1) of AB = 1×1 + 2×0 + 3×1 = 1 + 0 + 3 = 4. Element (1,2) of AB = 1×0 + 2×1 + 3×1 = 0 + 2 + 3 = 5. Element (2,1) of AB = 4×1 + 5×0 + 6×1 = 4 + 0 + 6 = 10. Element (2,2) of AB = 4×0 + 5×1 + 6×1 = 0 + 5 + 6 = 11.
Therefore AB = [[4, 5], [10, 11]].
Example 3: Expressing a matrix as sum of symmetric and skew-symmetric parts
Express the matrix A = [[2, 4], [6, 8]] as the sum of a symmetric and a skew-symmetric matrix.
Solution
First find A′ by interchanging rows and columns: A′ = [[2, 6], [4, 8]].
Symmetric part S = (1/2)(A + A′) = (1/2)[[2+2, 4+6], [6+4, 8+8]] = (1/2)[[4, 10], [10, 16]] = [[2, 5], [5, 8]].
Skew-symmetric part K = (1/2)(A − A′) = (1/2)[[2−2, 4−6], [6−4, 8−8]] = (1/2)[[0, −2], [2, 0]] = [[0, −1], [1, 0]].
Check: S + K = [[2, 5], [5, 8]] + [[0, −1], [1, 0]] = [[2, 4], [6, 8]] = A. ✓
Also note S′ = S (symmetric) and K′ = −K (skew-symmetric).
Common mistakes
Multiplying matrices of incompatible orders → check that columns of the first matrix equal rows of the second before multiplying.
Assuming AB = BA → matrix multiplication is not commutative; always keep the order as given.
Adding matrices of different orders → addition and subtraction require identical orders; if orders differ, the operation is not defined.
Forgetting to reverse order when transposing a product → remember (AB)′ = B′A′, not A′B′.
Confusing a row matrix with a column matrix → a 1 × n matrix (row) and an n × 1 matrix (column) are different objects with different multiplication rules.
Quick revision
- Order m × n means m rows and n columns; total elements = m × n.
- Matrix addition needs same order; matrix multiplication needs (columns of first) = (rows of second).
- AB is generally not equal to BA; always respect the given order.
- Transpose swaps rows and columns; (AB)′ = B′A′.
- Every square matrix = symmetric part + skew-symmetric part.
- Identity matrix I leaves any compatible matrix unchanged under multiplication.