Matrix Operations

Matrix and Scalar Entries. A matrix consists of

rows and

columns, and is called an m-by-n matrix.

$\displaystyle A = \left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n... ...\vdots & & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]$

where $a_{ij}$ is called the (i,j)-entry and associated with the value at i-th row and j-th column. For example, $\left[\begin{array}{ccc} 1 & 2 & -1 \\ 0 & -5 & 3 \end{array}\right]$ is a $2\times 3$ matrix, and (1,3)-entry is $-1$

Addition and Multiplication. Given two $m\times n$ matrices and , we can define the sum as an $m\times n$ matrix. For example,

$\displaystyle \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & ... ..._{13} \\ a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23} \end{bmatrix}$

Given a matrix of any size, we can define the scalar multiple with scalar . For example,

$\displaystyle c \; \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{... ...s a_{13} \\ c\times a_{21} & c\times a_{22} & c\times a_{23} \end{bmatrix}$

Matrix multiplication. Given $l\times m$ matrix $A=[a_{ij}]$ and $m\times n$ matrix $B=[b_{ij}]$ , we can introduce the multiplication of $l\times n$ matrix with (i,j)-entry

$\displaystyle c_{ij} = \sum_{k=1}^m a_{ik} b_{kj}$

Suppose, for example, that we have a $2\times 3$ matrix

and a $3\times 2$ matrix

. Then we can compute the multiplication

of $2 \times 2$ matrix by

$\displaystyle \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} ... ... b_{11} & b_{12} \\ b_{21} & b_{22} \\ b_{31} & b_{32} \end{bmatrix}$

$\displaystyle = \begin{bmatrix} a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31... ...a_{23} b_{31} & a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} \end{bmatrix}$

Matlab/Octave. We can define a matrix in either of the following forms:

> A = [4 0 5; -1 3 2]

> A = [4 0 5

-1 3 2]

You can execute basic matrix operations as follows.

> 5 * A

> A + B

> A * B

EXAMPLE 1. Let $A = \left[\begin{array}{ccc} 4 & 0 & 5 \\ -1 & 3 & 2 \end{array}\right]$ , $B = \left[\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 5 & 7 \end{array}\right]$ , and $C = \left[\begin{array}{cc} 2 & 3 \\ 0 & 1 \end{array}\right]$ .

Compute .
Is the matrix sum defined?
Compute .
Compute .

EXAMPLE 2. Compute in each of the following:

$A = \left[\begin{array}{cc} 2 & 3 \\ 1 &-5 \end{array}\right]$ and $B = \left[\begin{array}{ccc} 4 & 3 & 6 \\ 1 &-2 & 3 \end{array}\right]$ .
$A = \left[\begin{array}{ccc} 2 & -5 & 0 \\ -1 & 3 &-4 \\ 6 & -8 &-7 \\ -3 & 0 & 9 \end{array}\right]$ and $B = \left[\begin{array}{cc} 4 &-6 \\ 7 & 1 \\ 3 & 2 \end{array}\right]$ .

Properties of matrix multiplication. An $n\times n$ matrix is called a square matrix. When the square matrices and are of the same size, both and are defined. The matrix multiplication is associative. That is, whenever or is defined. However, the matrix multiplication is not commutative, that is, $A B \neq B A$ in general.

EXAMPLE 3. Let $A = \left[\begin{array}{cc} 5 & 1 \\ 3 &-2 \end{array}\right]$ and $B = \left[\begin{array}{cc} 2 & 0 \\ 4 & 3 \end{array}\right]$ . Show that these matrices do not commute.