1. Matrices
Rules & Operations

2. What is a Matrix
matrix is a collection of numbers arranged into a fixed number of rows and columns known as its Order. Usually the numbers are real numbers. Here is an example of a matrix with three rows and three columns:

3. Description The top row is row 1. The leftmost column is column 1.
This matrix order is 3x3 matrix because it has three rows and three columns.
In describing matrices, the format is:
rows X columns

4. Elements
Each number that makes up a matrix is called an element of the matrix. The elements in a matrix have specific locations.
The upper left corner of the matrix is row 1 column 1. In the above matrix the element at row 1 col 1 is the value 1. The element at row 2 column 3 is the value 4.6

5. Dimensions
The numbers of rows and columns of a matrix are called its dimensions. Here is a matrix with three rows and two columns:

6. Matrix Addition
If two matrices have the same number of rows and same number of columns, then the matrix sum can be computed:
Below we have a 3 x 2 matrix added to a 3 x 2 matrix:

7. Addition Practice

8. Matrix Subtraction
If A and B have the same number of rows and columns, then A - B is defined as A + (-B). Usually you think of this as:

9. Multiplication of a Matrix by a Scalar
A matrix can be multiplied by a scalar (by a real number) as follows:
To multiply a matrix by a scalar, multiply each element of the matrix by the scalar.
Here is an example of this. (In this example, the variable a is a scalar.)

10. Matrix Multiplication
We can only multiply matrices if the number of columns in the first matrix is the same as the number of rows in the second matrix.
Multiplying a 2 × 3 matrix by a 3 × 4 matrix is possible and it gives a 2 × 4 matrix as the answer.
Multiplying a 7 × 1 matrix by a 1 × 2 matrix is okay; it gives a 7 × 2 matrix
A 4 × 3 matrix times a 2 × 3 matrix is NOT possible.

11. Matrix Multiplication
As an example, let's take a general 2 × 3 matrix multiplied by a 3 × 2 matrix.
The answer will be a 2 × 2 matrix.

12. Matrix Multiplication
We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a11 (top left) of the answer matrix.

13. Matrix Multiplication
We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a12.

14. Matrix Multiplication
Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a21.

15. Matrix Multiplication
Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a22.

16. Matrix Multiplication
So the result of multiplying our 2 matrices is as follows:

17. Matrix Multiplication
Multiply:

18. Answer
This is 2×3 times 3×2, which will give us a 2×2 answer

19. Multiply

20. Answer