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Chapter 17 LA3: Matrix Algebra

We introduced matrices when discussing systems of linear equations. Recall that an \(m \times n\) matrix is a rectangular array of numbers, written in the form

\begin{equation*} \begin{pmatrix} a_{11} \amp a_{12} \amp \cdots \amp a_{1n}\\ a_{21} \amp a_{22} \amp \cdots \amp a_{2n}\\ \vdots \amp \vdots \amp \ddots \amp \vdots\\ a_{m1} \amp a_{m2} \amp \cdots \amp a_{mn}\\ \end{pmatrix}, \end{equation*}

which is sometimes written in the abbreviated form

\begin{equation*} A=\left(a_{ij}\right)_{m \times n}. \end{equation*}

Note that we usually use a capital letter to denote a matrix.

In the context of systems of linear equations the main concept was row reducing a matrix to produce equivalent matrices. In different contexts, however, other operations on matrices have proved useful and we shall discuss these operations below. Before doing this, though, we will introduce some new terminology.

Definition 17.0.1.

Two matrices are said to be equal if they are of the same size and all corresponding entries are equal.

Let \(A = \begin{pmatrix}5 \amp 0 \amp 1\\ -3 \amp \sqrt{2} \amp \frac{1}{2}\end{pmatrix}, \quad B = \begin{pmatrix}5 \amp -3 \\ 0 \amp \sqrt{2} \\ 1 \amp \frac{1}{2}\end{pmatrix}, \quad C = \begin{pmatrix} 5 \amp 0 \amp 1\\ -3 \amp a \amp b\end{pmatrix}.\)

  1. For what values of \(a\) and \(b\) does \(A=C\text{?}\)

  2. For what values of \(a\) and \(b\) does \(B=C\text{?}\)

Answer.

  1. \(a=\sqrt{2}\) and \(b=\dfrac{1}{2}\)

  2. None.

Solution.

  1. Since \(A\) and \(C\) are of the same size, the matrices will be equal when \(a=\sqrt{2}\) and \(b=\dfrac{1}{2}\text{.}\)

  2. Since \(B\) and \(C\) are not the same size they can never be equal.

Definition 17.0.3.

The zero \(m \times n\) matrix is the \(m \times n\) 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*}

Note that by using the symbol \(0\) to denote the zero matrix we have to tell from the context whether the \(0\) refers to a number or a matrix.

Definition 17.0.4.

For the matrix \(A=(a_{ij})_{m \times n}\text{,}\) its transpose, denoted by \(A^T\text{,}\) is defined by \(A^T = \left(a_{ji}\right)_{n \times m}\text{.}\)

Note that this definition is saying that the rows of matrix \(A\) are the columns of its transpose \(A^T\) and vice versa.

Let

\begin{equation*} B = \begin{pmatrix}5 \amp -3 \\ 0 \amp \sqrt{2} \\ 1 \amp \frac{1}{2}\end{pmatrix}, \quad C = \begin{pmatrix} 5 \amp 0 \amp 1\\ -3 \amp a \amp b\end{pmatrix}. \end{equation*}

For what values of \(a\) and \(b\) does \(B^T=C\text{?}\)

Answer.
\(a=\sqrt{2}, \ b=\dfrac{1}{2}\)
Solution.
Since \(B^T = \begin{pmatrix} 5 \amp 0 \amp 1\\ -3 \amp \sqrt{2} \amp \frac{1}{2}\end{pmatrix}\text{,}\) \(B^T = C\) when \(a=\sqrt{2}\) and \(b=\frac{1}{2}\text{.}\)

Definition 17.0.6.

A square matrix is a matrix that has the same number of rows and columns.

Matrix \(A\) is a square matrix whereas matrix \(B\) is not.

\begin{equation*} A = \begin{pmatrix}-1 \amp 3 \amp 4\\ 5 \amp 0 \amp -2 \\ 3 \amp \frac{1}{3} \amp 6 \end{pmatrix}, \quad B = \begin{pmatrix} -1 \amp 3 \\ 5 \amp 0 \\ 3 \amp \frac{1}{3}\end{pmatrix} \end{equation*}

Definition 17.0.8.

The \(n \times n\) identity matrix, \(I_n\), is the matrix with \(n\) rows and \(n\) columns with all entries on the main diagonal equal to \(1\) and all other entries \(0\text{,}\) i.e.
\begin{equation*} I_n = \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*}

Often, when the size of the identity matrix is clear the subscript is dropped from the notation and the identity matrix is denoted just by \(I\text{.}\)