01Zmatrices


 * Operations with Matrices **

Matrices are a way of grouping numbers and operations together. They have many applications. For example they are used in computer graphics to project a 3-dimensional view onto a 2-dimensional screen. They can also be used to solve simultaneous linear equations (sometimes called systems of linear equations).

A ** matrix ** is an array of numbers surrounded by square brackets [ ]

math \left[ \begin{matrix} 2&1&5 \\ 3&0&-2 \\ \end{matrix} \right] math

The dimensions of a matrix are the number of rows by the number of columns. So a 2 by 3 matrix has 2 rows and 3 columns.

A matrix that has only one row or only one column is often called a ** vector **.

Adding Matrices

Two matrices with the same dimensions can be added by adding the values in corresponding positions.

math \left[ \begin{matrix} 1&3 \\ 2&8 \\ \end{matrix} \right] + \left[ \begin{matrix} 3&-5 \\ 2&-2 \\ \end{matrix} \right] = \left[ \begin{matrix} 4&-2 \\ 4&6 \\ \end{matrix} \right] math

ie in the first row and first column of the two original matrices, we have 1 & 3 So the first row and first column of the solution matrix is 1 + 3 = 4

If the matrices have different dimensions, they cannot be added.

Matrices on the Classpad

To enter a matrix onto the classpad, go to the virtual keyboard.

Go to the **2D tab** and select the **CALC** option.

The top row of icons gives you the different matrices.

Tap the icon twice (or more) to get a larger matrix.

Tap combinations of the three icons to get non-square matrices.

Multiplying a matrix by a constant

Each value inside the matrix is multiplied by the constant

math \textbf{Eg 1: } \; 4 \left[ 1 \quad 2 \quad 3 \right] = \left[ 4 \quad 8 \quad 12 \right] math

math \textbf{Eg 2: } \; 3 \left[ \begin{matrix} a&b \\ c&d \\ \end{matrix} \right] = \left[ \begin{matrix} 3a&3b \\ 3c&3d \\ \end{matrix} \right] math

Multiplying Two Matrices
 * Each row in the first matrix is multiplied by each column in the second matrix.
 * The matching values are multiplied together and the results added.
 * The position of the final result in the answer matrix is given by the row number from the first matrix and the column number from the second matrix.

math \left[ \begin{matrix} a&b \\ c&d \\ \end{matrix} \right] \times \left[ \begin{matrix} w&x \\ y&z \\ \end{matrix} \right] = \left[ \begin{matrix} aw+by&.. \\ ..&.. \\ \end{matrix} \right] math

In this demonstration,
 * the first row in the first matrix is [a b],
 * the first column in the second matrix is [w y]
 * multiply the values in matching positions, then add the result gives aw + by
 * the result goes in row 1, column 1 of the answer matrix

Repeat this for the other three positions of the answer matrix and you get:

math \left[ \begin{matrix} a&b \\ c&d \\ \end{matrix} \right] \times \left[ \begin{matrix} w&x \\ y&z \\ \end{matrix} \right] = \left[ \begin{matrix} aw+by&ax+bz \\ cw+dy&cx+dz \\ \end{matrix} \right] math


 * IMPORTANT! ** The order of multiplication is important. In matrices A × B is rarely equal to B × A

Note: If the number of columns in the first matrix does not match the number of rows in the second matrix, the two matrices cannot be multiplied.

Example 1

math \left[ \begin{matrix} 1&3 \\ 4&2 \\ \end{matrix} \right] \times \left[ \begin{matrix} -1&2 \\ 3&1 \\ \end{matrix} \right] = \left[ \begin{matrix} 1 \times -1 + 3 \times 3&2+3 \\ -4+6&8+2 \\ \end{matrix} \right] = \left[ \begin{matrix} 8&5 \\ 2&10 \\ \end{matrix} \right] math

Example 2

math \left[ \begin{matrix} 4&-2 \\ 3&1 \\ \end{matrix} \right] \times \left[ \begin{matrix} 3 \\ 2 \\ \end{matrix} \right] = \left[ \begin{matrix} 4 \times 3 + -2 \times 2 \\ 9+2 \\ \end{matrix} \right] = \left[ \begin{matrix} 8 \\ 11 \\ \end{matrix} \right] math

Example 3

math \left[ \begin{matrix} 2&5 \\ 3&4 \\ \end{matrix} \right] \times \left[ \begin{matrix} x \\ y \\ \end{matrix} \right] = \left[ \begin{matrix} 2x+5y \\ 3x+4y \\ \end{matrix} \right] math

For another site that demonstrates this idea, go here: MathsIsFun

Dividing Matrices

To divide two matrices, you will need to take the Inverse of the divisor and then multiply.

Go to Inverse Matrices

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