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2. Matrices
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Algebra 2

2.01 Introduction to matrices

Lesson
Practice
Lesson

Concept summary

Mathematical information can be represented in a variety of ways. A matrix is a rectangular grouping of numbers that can be used to organize and display data. In general, matrices are typically notated by a capital letter.

Matrix

A rectangular array of numbers or variables

Example:

M = \begin{bmatrix} a & 2 \\ 3 & d \end{bmatrix}

The dimensions of a matrix are referred to as the order and are always listed by row then column in the form r\times c where r represents the number of rows and c represents the number of columns. Special forms of a matrix can be classified by their order:

Square matrix

A matrix with order n\times n

Example:

3 \times 3 matrix: \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}

Row matrix

A matrix with order 1\times n

Example:

1 \times 3 matrix:\text{ } \\ \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}

Column matrix

A matrix with order n\times 1

Example:

3 \times 1 matrix:\text{} \\ \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}

The elements of a matrix are the individual numbers and variables and can be identified by their position using subscript notation a_{mn} where m represents the row the element is located in and n represents the column as shown in the matrix below:

\displaystyle A=\begin{bmatrix} a_{11} & a_{12} & \ldots & a_{1m} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \ldots & a_{mn} \end{bmatrix}
\bm{a_{ij}}
is the element in row i and column j
\bm{m}
is the number of rows in matrix A
\bm{n}
is the number of columns in matrix A

Matrices can only be equal if they have the same order and every element within each matrix is equal. For example, if A=B, then a_{11}=b_{11}, a_{12}=b_{12}... etc.

Worked examples

Example 1

Consider the following table that displays the favorite color of students by grade level:

6th grade7th grade8th grade
Red553
Blue865
Black247
a

Represent the data in the table as a matrix, A.

Approach

Each element in the table should have a corresponding element in the matrix and generally, the layout of the matrix will match the layout of the table.

Solution

A = \begin{bmatrix} 5 & 5 & 3 \\ 8 & 6 & 5 \\ 2 & 4 & 7 \end{bmatrix}

Reflection

Pay attention to the relative position of each element as the row and column it's in each have meaning in context of the data

b

Determine the order of the matrix.

Approach

The order of a matrix is written as r\times c where r represents the number of rows and c represents the number of columns.

Solution

The order of matrix A is 3\times 3.

c

Identify the value of a_{23} and interpret its meaning in context.

Approach

The subscript indicates the position of the element by row and column respectively.

Solution

a_{23} represents the element in the 2nd row and 3rd column.

a_{23}=5 and represents the number of 8th grade students whose favorite color is blue.

Example 2

Consider the matrix equation: \begin{bmatrix} x & 5 \\ 9 & 6-y \end{bmatrix}=\begin{bmatrix} 3 & 5 \\ 9 & 2 \end{bmatrix}

a

Solve for x.

Approach

Two matrices are equal if each of their corresponding elements are equal. To solve, we'll identify the element corresponding to x and write an equality statement.

Solution

The variable x is in the first row and first column of \begin{bmatrix} x & 5 \\ 9 & 6-y \end{bmatrix} and 3 is in the first row and first column of \begin{bmatrix} 3 & 5 \\ 9 & 2 \end{bmatrix}. Since the matrices are equal we have: x=3

b

Solve for y.

Approach

Two matrices are equal if each of their corresponding elements are equal. To solve, we'll identify the element containing y and set it equal to its corresponding element in the equal matrix.

Solution

The variable y is in the second row and second column of \begin{bmatrix} x & 5 \\ 9 & 6-y \end{bmatrix} and 2 is in the second row and second column of \begin{bmatrix} 3 & 5 \\ 9 & 2 \end{bmatrix}. Since the matrices are equal we have:

\displaystyle 6-y\displaystyle =\displaystyle 2If the matrices are equal, then the corresponding elements are equal
\displaystyle 6\displaystyle =\displaystyle 2+yAdd y to both sides
\displaystyle 4\displaystyle =\displaystyle ySubtract 2 from both sides

Therefore, y=4.

Outcomes

A2.N.M.A.1

Use matrices to represent data in a real-world context. Interpret rows, columns, and dimensions of matrices in terms of the context.*

A2.MP1

Make sense of problems and persevere in solving them.

A2.MP4

Model with mathematics.

A2.MP6

Attend to precision.

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