Skip to main content

Chapter 18 LA4: Square Matrices

Recall that a square matrix is a matrix with the same number of rows as columns. We call an \(n\times n \) matrix a square matrix of order \(n \text{.}\) When we add or multiply two square matrices of order \(n \) we always obtain a square matrix of order \(n \text{.}\) The zero matrix, \(0 \text{,}\) of order \(n \) is the matrix with all entries \(0 \text{,}\) i.e.

\begin{equation*} 0 = \begin{pmatrix} 0 \amp 0 \amp \cdots \amp 0 \\ 0 \amp 0 \amp \cdots \amp 0 \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ 0 \amp 0 \amp \cdots \amp 0 \end{pmatrix} \end{equation*}

and has the properties

  1. \(\displaystyle 0A=A0=0 \)

  2. \(\displaystyle A+0=A \)

  3. \(\displaystyle A-A=0 \)

where \(A \) is any square matrix of order \(n \text{.}\) The identity matrix, \(I, \) of order \(n \) is the \(n\times n \) matrix with \(1\text{'s} \) on the main diagonal and all other entries \(0 \text{.,}\) i.e.

\begin{equation*} I = \begin{pmatrix} 1 \amp 0 \amp \cdots \amp 0 \\ 0 \amp 1 \amp \cdots \amp 0 \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ 0 \amp 0 \amp \cdots \amp 1 \end{pmatrix} \end{equation*}

The identity matrix has the property that for any square matrix of order \(n, \) \(A, \)

  • \(\displaystyle IA=AI=A \)

\(I \) is the only matrix that satisfies this property.