Algebra 2 by Richard Wright
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Objectives:
SDA NAD Content Standards (2018): AII.4.1
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.
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.
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.
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
You can only add and subtract matrices that are the same dimensions.
The matrices must be the same dimension.
To add or subtract matrices,
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.
$$ \left[\begin{matrix}5+1&7+\left(-3\right)\\0+2&-2+\left(-5\right)\end{matrix}\right] $$
$$ =\left[\begin{matrix}6&4\\2&-7\end{matrix}\right] $$
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.
$$ \left[\begin{matrix}1-4+\left(-4\right)\\-2-\left(-7\right)+0\end{matrix}\right] $$
$$ =\left[\begin{matrix}-7\\5\end{matrix}\right] $$
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.
A scalar is a number by itself. So, multiplying a matrix by a scalar is a number multiplied with a matrix like \(2\left[\begin{matrix}0&1\\3&4\end{matrix}\right]\).
To multiply a scalar with a matrix,
Simplify \(5\left[\begin{matrix}2&7&-4\\1&0&-3\end{matrix}\right]\).
Solution
Distribute the scalar, 5, to all the elements.
$$ =\left[\begin{matrix}5\left(2\right)&5\left(7\right)&5\left(-4\right)\\5\left(1\right)&5\left(0\right)&5\left(-3\right)\end{matrix}\right] $$
$$ =\left[\begin{matrix}10&35&-20\\5&0&-15\end{matrix}\right] $$
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.
| June | Benton Harbor | South Bend |
|---|---|---|
| Precip Days | 15 | 18 |
| Clear Days | 11 | 12 |
| Ab Norm T | 5 | 7 |
| July | Benton Harbor | South Bend |
|---|---|---|
| Precip Days | 12 | 11 |
| Clear Days | 15 | 19 |
| Ab Norm T | 3 | 6 |
Solution
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.
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Mixed Review