Equations involving matrices
Two matrices are said to be equal if every corresponding elements in the matrices are equal. \begin{bmatrix} 3 & 4\\ -2 & 7 \end{bmatrix} = \begin{bmatrix} m & 4\\ -2 & n \end{bmatrix}
So in this case, since the two matrices are equal, then m=3 and n=7.
Simultaneous equations follow the convention of organising the coefficients into matrix notation. For instance, consider the two equations x+2y=7 and 2x-5y=-4. This system can be written like this using matrices. \begin{bmatrix} 1 & 2\\ 2 & -5 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 7\\ -4 \end{bmatrix}
This will be covered in more detail when looking at matrix multiplication in a later lesson.
Examples
Example 1
Consider the equation \begin{bmatrix} x-5 \\ -4y \end{bmatrix} = \begin{bmatrix} -7 \\ -16 \end{bmatrix}.
Solve for x.
Solve for y.
Two matrices are said to be equal if every corresponding elements in the matrices are equal.
Use of matrices to model practical examples
When simply being used to store data, is it not necessary to use the labelling conventions and simply create the matrix. Consider the following table on distances between towns:
| A | B | C | |
|---|---|---|---|
| \text{A} | 0 | 23 | 17 |
| \text{B} | 23 | 0 | 43 |
| \text{C} | 17 | 43 | 0 |
The above situation is converted to a matrix like this: \begin{bmatrix} 0& 23 & 17 \\ 23 & 0& 43 \\ 17 & 43 &0 \end{bmatrix}
However, when undertaking mathematical processes using the data stored in matrices, the defining of rows and columns must follow normal conventions.
Two-way frequency tables follow the convention of independent variable headings in the columns and dependent variables as the rows.
A survey was completed at a school that has both secondary and primary school students. They were asked if the playground should have fewer, same or more seating options. The results are displayed in the table below:
| Attitude | Primary | Secondary |
|---|---|---|
| \text{Fewer} | 5 | 2 |
| \text{Same} | 29 | 9 |
| \text{More} | 33 | 36 |
| \text{Total} | 67 | 47 |
This can be written as a matrix. The result would be: \begin{bmatrix} 5&2\\ 29&9\\ 33&36\\ 67&47 \end{bmatrix}
Examples
Example 2
Jack, a chef, is known for his CrazyCookie, which requires 360 g of yeast, 410 g of salt, 340 g of flour, 230 g of sugar and 120 g of honey. He is also known for his ScrumptiousSurprise, which requires 420 g of yeast, 390 g of salt, 330 g of flour, 200 g of sugar and 80 g of honey.
Organise the data into a 2 \times 5 matrix.
Let the first row be the values for CrazyCookie and the second row be the values for ScrumptiousSurprise.
Let the columns be values of each ingredient, in the order given in the instructions.
Matrices can be used for storing and displaying information in a variety of applications, such as tables and lists of data and networks.
The order that you put the data in your matrix is important. You must decide what each column and row represents.