Algebra 2 by Richard Wright

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1-04 Perform Basic Matrix Operations (12.1)

Mr. Wright teaches the lesson.

Objectives:

SDA NAD Content Standards (2018): AII.4.1

tables
Figure 1: Tables. (Pixabay/StockSnap)

A table is a good way to organize data such as dates. The calendar on the picture arranges the dates into a rectangular array of numbers. This is a matrix. This lesson address what a matrix is and how to do basic matrix operations.

Matrices

A matrix is a rectangular arrangement of numbers. In computing they are often called arrays. Matrices are a convenient way to organize and work with a lot of data. Matrices can also be used to solve systems of equations.

Dimensions and Equality

Consider the matrix \(\left[\begin{matrix}2&-1&5&a\\2&y&6&b\\3&14&x&c\end{matrix}\right]\). How could it be described. The size of the matrix is called the dimension or order. The dimension is the number of rows by the number of columns. A row is a horizontal list of numbers, and a column is a vertical list. The above matrix has dimension 3 × 4 because it has 3 horizontal row and 4 vertical columns.

In order for two matrices to be equal, they must be the same dimensions and corresponding elements must be the same. For example, \(\left[\begin{matrix}1&3\\2&4\end{matrix}\right]=\left[\begin{matrix}1&3\\2&4\end{matrix}\right]\) are equal matrices.

Example 1

Find the value of the variables \(\left[\begin{matrix}5&2y\\x+1&4\end{matrix}\right]=\left[\begin{matrix}w&-6\\7&z-2\end{matrix}\right]\).

Solution

Set the corresponding elements equal to each other.

The top left elements.

5 = w

The top right elements.

2y = −6

y = −3

The bottom left elements.

x + 1 = 7

x = 6

The bottom right elements.

4 = z − 2

z = 6

Adding and Subtracting Matrices

You can only add and subtract matrices that are the same dimensions.

Add and Subtract Matrices

The matrices must be the same dimension.

To add or subtract matrices,

  1. Add or subtract the corresponding elements.

Example 2

Simplify \(\left[\begin{matrix}5&7\\0&-2\end{matrix}\right]+\left[\begin{matrix}1&-3\\2&-5\end{matrix}\right]\).

Solution

Add the corresponding elements: top left with top left, top right with top right, etc.

$$ \left[\begin{matrix}5+1&7+\left(-3\right)\\0+2&-2+\left(-5\right)\end{matrix}\right] $$

$$ =\left[\begin{matrix}6&4\\2&-7\end{matrix}\right] $$

Example 3

Simplify \(\left[\begin{matrix}1\\-2\end{matrix}\right]-\left[\begin{matrix}5\\-7\end{matrix}\right]+\left[\begin{matrix}-4\\0\end{matrix}\right]\).

Solution

Add or subtract the corresponding elements.

$$ \left[\begin{matrix}1-4+\left(-4\right)\\-2-\left(-7\right)+0\end{matrix}\right] $$

$$ =\left[\begin{matrix}-7\\5\end{matrix}\right] $$

Example 4

Simplify \(\left[\begin{matrix}9&13\\-1&6\end{matrix}\right]-\left[\begin{matrix}17&-2\\11&10\\-5&6\end{matrix}\right]\).

Solution

You cannot add because the matrices are different dimensions.

Scalar Multiplication

A scalar is a number by itself. So, multiplying a matrix by a scalar is a number multiplied with a matrix like \(2\left[\begin{matrix}0&1\\3&4\end{matrix}\right]\).

Scalar Multiplication

To multiply a scalar with a matrix,

  1. Multiply each element by the scalar like you are distributing.

Example 5

Simplify \(5\left[\begin{matrix}2&7&-4\\1&0&-3\end{matrix}\right]\).

Solution

Distribute the scalar, 5, to all the elements.

$$ =\left[\begin{matrix}5\left(2\right)&5\left(7\right)&5\left(-4\right)\\5\left(1\right)&5\left(0\right)&5\left(-3\right)\end{matrix}\right] $$

$$ =\left[\begin{matrix}10&35&-20\\5&0&-15\end{matrix}\right] $$

Problem Solving

Example 6

The National Weather Service keeps track of weather. The following table shows the number of days with precipitation, days with no clouds, and days with above normal temperatures for two months.

JuneBenton HarborSouth Bend
Precip Days1518
Clear Days1112
Ab Norm T57

JulyBenton HarborSouth Bend
Precip Days1211
Clear Days1519
Ab Norm T36
  1. Represent each table as matrices.
  2. What is meaning of the first matrix + second matrix?
  3. Use matrix operations to find the total weather stats of each city.

Solution

  1. \(\left[\begin{matrix}15&18\\11&12\\5&7\end{matrix}\right]\) and \(\left[\begin{matrix}12&11\\15&19\\3&6\end{matrix}\right]\)
  2. The total number of days of each type for each city for June and July combined.
  3. Add the matrices together.

    $$ \left[\begin{matrix}15+12&18+11\\11+15&12+19\\5+7&3+6\end{matrix}\right] $$

    $$ =\left[\begin{matrix}27&29\\26&31\\12&9\end{matrix}\right] $$

    Benton Harbor had 27 precip days, 26 clear days, and 12 above normal temperature days. South Bend had 29 precip days, 31 clear days, and 9 above normal temperature days.

Practice Problems

Page 650 #1, 5, 9, 13, 15, 17, 19, 21, 23, 25, 29, 33, 35, 37, 39, and Mixed Review = 20

    Mixed Review

  1. (1-03) Solve \(\left\{\begin{alignat}{4} 2x&+&y&-&3z&=&2 \\ -2x&-&y&+&z&=&-4 \\ 2x&+&2y&-&2z&=&6 \end{alignat}\right.\).
  2. (1-03) Solve \(\left\{\begin{alignat}{4} x&+&y&+&z&=&4 \\ &&y&+&2z&=&5 \\ &&&&z&=&3 \end{alignat}\right.\).
  3. (1-02) Solve algebraically \(\left\{\begin{alignat}{3} 2x&-&y&=&5 \\ x&+&3y&=&-8 \end{alignat}\right.\).
  4. (1-01) Solve by graphing \(\left\{\begin{align} y&=2x-5 \\ y&=-x+1 \end{align}\right.\).
  5. (0-03) Solve 2|x + 1| = 4.

Answers

  1. (1, 3, 1)
  2. (2, −1, 3)
  3. (1, −3)
  4. (2, −1)
  5. −3, 1