Higher Mathematics 1st Paper · Classes 11-12 · Chapter 1 · 1.1

Matrices and Their Types.

Picture a result sheet: marks laid out in rows and columns so the eye finds any one at once. Mathematics gives that tidy arrangement a name and a set of rules. This note covers the name, how to state a matrix's order, and the first kinds of matrix you must be able to recognise.

1. A matrix and its entries

Data in science and mathematics are often laid out in a rectangle, along rows (horizontal lines) and columns (vertical lines). That rectangular arrangement is called a matrix: a rectangular array of numbers enclosed in brackets.

Each number inside a matrix is an entry. For the brackets, ( ), [ ] or ‖ ‖ will all do. For names, capital letters A, B, C ... stand for matrices and small letters a, b, c ... stand for entries.

The horizontal lines of numbers are rows and the vertical lines are columns. Take a matrix whose rows are (6, 4, 8), (1, 3, 5), (2, 7, 9). Its 1st row is 6, 4, 8 and its 2nd column is 4, 3, 7.

2. Order and the general form

A matrix with m rows and n columns has order m × n, read “m by n”. For example the matrix with rows (0, 0, 2) and (0, 3, 0) has order 2 × 3. Once you know the order, counting entries is easy: total entries = number of rows × number of columns, so a 2 × 3 matrix has 6 entries.

In general form a matrix A with m rows and n columns is written A = [a_ij]_m×n, or (a_ij)_m×n, or ‖a_ij‖_m×n. Here a_ij is the entry in the i-th row and j-th column, with i running from 1 to m and j from 1 to n. Use the calculator below: type the order and the number of entries yourself and see whether they match.

3. The first few kinds

A matrix with only one row is a row matrix, such as (4, 1, 3), of order 1 × 3. A matrix with only one column is a column matrix, such as the column 2, 5, 6, of order 3 × 1.

When the number of rows equals the number of columns (m = n) the matrix is a square matrix, and n is called its order. For example the matrix with rows (4, 1, 2), (0, 3, 2), (1, −2, 3) is a square matrix of order 3 × 3.

In a square matrix A = (a_ij)_n×n the diagonal running through the entries a₁₁, a₂₂, ..., a_nn is the principal (main) diagonal. The next few kinds are all defined around it.

If every entry below the principal diagonal is zero (a_ij = 0 when i > j) the matrix is upper triangular. For example, rows (2, 1, −1), (0, 2, 1), (0, 0, −3).

The other way round, if every entry above the diagonal is zero (a_ij = 0 when i < j) the matrix is lower triangular. For example, rows (3, 0, 0), (−4, −1, 0), (4, 2, −3).

4. From diagonal to identity, and the zero matrix

A square matrix with a_ij = 0 whenever i ≠ j, that is, with every entry off the diagonal zero, is a diagonal matrix. For example, 6, 7, 8 down the diagonal and 0 everywhere else.

If the non-zero entries of a diagonal matrix are all equal, it is a scalar matrix. For example, 5, 5, 5 down the diagonal.

A square matrix with a_ij = 0 (i ≠ j) and a_ij = 1 (i = j) is the unit or identity matrix, written I_n. For instance I₂ has rows (1, 0), (0, 1). Useful later: |I| = 1, Iᵀ = I and I² = I.

A matrix whose entries are all zero is a zero (null) matrix. It need not be square: a 2 × 3 matrix of zeros is a zero matrix too.

Now take eight matrices one at a time. Tap every kind that fits, then press check.

The kinds nest inside one another. With p, q, r on the diagonal and 0 elsewhere, the matrix is diagonal. When p = q = r it is scalar, and when p = q = r = 1 it is the identity. So the matrix with 3, 3, 3 on the diagonal is both diagonal and scalar, but not an identity. Move the handles and see.

5. Check yourself

The first question is in the exam's style.

In the second, think about how the kinds relate.

For the last, recall the definition.

The whole note on seven cards.

Source: Higher Mathematics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Matrices and Determinants”, section 1.1.