M1: Matrices – Introduction

This module discusses matrices, their order, row and column matrices, square matrices and the identity matrix.

• A matrix is a rectangular array of elements.
• Matrices are usually denoted by upper case letters.
• The elements are usually written within brackets.
• The order or shape of the matrix is determined by the number of rows and columns of the matrix.
• The number of rows is always given first then the number of columns. Example. \[\begin{align*} A & =\left[\begin{array}{ccc} 1 & 2 & -9\\ 2 & 5 & -3 \end{array}\right] \end{align*}\]

\(A\) has 2 rows and 3 columns and is called a \(2\times3\) matrix.1 This is verbally stated as a 2 by 3 matrix.

A matrix with \(m\) rows and \(n\) columns is called a matrix of order \(m\times n\).2 This is verbally termed an “m by n matrix”.

Square matrix

A matrix with the same number of rows and columns is called a square matrix.

Example: \[\begin{align*} B & =\left[\begin{array}{cc} 2 & 3\\ 2 & 5 \end{array}\right] \end{align*}\]

\(B\) is a square \(2\times2\) matrix

Unit matrix

A unit (or identity) matrix is a square matrix with diagonal elements equal to one, and all other elements equal to zero. The unit matrix is usually denoted by \(I\).

\(I_{3}\) is a \(3\times3\) unit matrix

Example: \[\begin{align*} I_{3} & =\left[\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right] \end{align*}\]

Row matrix

A matrix with one row is called a row matrix.

Example: \[\begin{align*} D & =\left[\begin{array}{cccc} 2 & 1 & 0 & 4\end{array}\right] \end{align*}\] is a \(1\times4\) row matrix

Column matrix

A matrix with one column is called a column matrix.

Example: \[\begin{align*} E & =\left[\begin{array}{c} 2\\ -4\\ 1 \end{array}\right] \end{align*}\]

is a \(3\times1\) column matrix

Zero matrix

A zero matrix has all elements equal to zero. A zero matrix can be written as \(0\).

Example: \[\begin{align*} 0 & =\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right] \end{align*}\]

is a \(2\times2\) zero matrix

Equal matrices

For two matrices to be equal, they must be the same shape and the corresponding elements must be equal.

If \(A\) equals \(B\) then \[\begin{align*} A & =\left[\begin{array}{ccc} 2 & 5 & b\\ 5 & 3 & 1\\ 2 & 0 & -2 \end{array}\right] & B & =\left[\begin{array}{ccc} 2 & 5 & 7\\ 5 & a & 1\\ 2 & 0 & -2 \end{array}\right] \end{align*}\]

If \(A\) and \(B\) are equal then \(a\) = 3 and \(b\) = 7.

Exercise

1. Write down the order of the following matrices.

1. \(\left[\begin{array}{ccc} 7 & -5 & 0\\ 6 & 2 & -1 \end{array}\right]\)
2. \(\left[\begin{array}{cc} 0 & 2\\ 1 & 1 \end{array}\right]\)
3. \(\left[\begin{array}{c} 2\\ -4\\ 1\\ 1 \end{array}\right]\)
4. \(\left[\begin{array}{cc} 1 & 1\\ 3 & 0\\ -2 & 3 \end{array}\right]\)

2. Which of the following matrices are equal?

\[\begin{align*} A & =\left[\begin{array}{cc} 3 & 0\\ 1 & -2 \end{array}\right] & B & =\left[\begin{array}{cc} 3 & 1\end{array}\right] & C & =\left[\begin{array}{cc} 3 & 0\end{array}\right] & D & =\left[\begin{array}{cc} 3 & 0\\ 1 & -2 \end{array}\right] \end{align*}\]

\[\begin{align*} E & =\left[\begin{array}{ccc} 3 & 5 & 1\\ 2 & 0 & 1 \end{array}\right] & F & =\left[\begin{array}{cc} 0 & 3\end{array}\right] & G & =\left[\begin{array}{ccc} 3 & 5 & 1\\ 2 & 0 & 1\\ 1 & 3 & 0 \end{array}\right] \end{align*}\]