Refresher - Matrices - Basics-2

On a practical level, we can use a matrix to display data, such as comparing the price of gasoline in various cities.

	Regular	Super	Premium
LA	$3.00	$3.10	$3.20
GG	$2.90	$3.00	$3.10
SD	$3.10	$3.20	$3.30

Elements of a matrix could also be terms of a matrix equation:

$[\begin{matrix} 1 & - 3 \\ - 3 & 1 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 2 \end{matrix}]$

This is another way to represent the system

${\begin{cases} x = 3 y \\ y = 3 x + 2 \end{cases}$

A matrix is a rectangular array of numbers, which are called elements. If a matrix has m rows and n columns, then the dimensions m by n are written m x n. In a square matrix, the number of rowsand columns are equal. Two matrices are equal if their corresponding elements are equal.

Formula	Description
Addition	To add matrices of the same dimension, just add the corresponding elements.
Subtraction	To subtract matrices of the same dimension, just subtract the corresponding elements.
Identity	Identity Matrix, /, is a square matrix with the leading diagonal elements assigned 1 and the other elements assigned 0. The examples are a 2x2 identity matrix and a 3x3 identity matrix.
Product of scale and matrix	To find the product of a scalar and a matrix, just multiply each element by the scalar
Product of two matrices	To find the product of two matrices, we must first determine if a product exists by looking at the dimensions. The product of a mxn matrix and a nxt matrix is a mxt matrix. In other words, the number of columns in the first matrix must match the number of rows in the second matrix in order for a product to exist.

Once we determine that multiplication is possible, we multiply the rows and columns.

Let $A = [\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}] and C = [\begin{matrix} c_{1} & d_{1} \\ c_{2} & d_{2} \end{matrix}]$

Each element of the product of AC is the product of one row of matrix A and one column of matrix C. This is also the dot product of one row and one column.

$A C = [\begin{matrix} a_{1} c_{1} + b_{1} c_{2} & a_{1} d_{1} + b_{1} d_{2} \\ a_{2} c_{1} + b_{2} c_{2} & a_{2} d_{1} + b_{2} d_{2} \end{matrix}]$ or $[\begin{matrix} Row 1 \cdot Col 1 & Row 1 \cdot Col 2 \\ Row 2 \cdot Col 1 & Row 2 \cdot Col 2 \end{matrix}]$

When multiplying matrices, it is important to realize that the order of the multiplicands is significant, in other words [A][B] is not necessarily equal to [B][A]. In other words, matrix multiplication is not commutative.

$A B = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}] = [\begin{matrix} 5 & 6 \\ 11 & 12 \end{matrix}]$
$B A = [\begin{matrix} 1 & 0 \\ 2 & 3 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] = [\begin{matrix} 1 & 2 \\ 11 & 16 \end{matrix}]$
$A B \neq B A$

For more on matrix operations, click here Links to an external site..

Some matrices have multiplicative inverses. If $A = [\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}]$ , the inverse is denoted A-1. In order to be inverses, $A • A^{- 1} = I$ ,where I is the identity matrix.

For an applet that illustrates inverse operations, click here Links to an external site..