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Matrices
Rules & Operations
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:
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

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
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Dimensions

The numbers of rows and columns of a matrix
are called its dimensions. Here is a matrix with
three rows and two columns:
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:

Addition Practice
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:
Multiplication of a Matrix by a
Scalar

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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.)
Matrix Multiplication

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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.
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.
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.
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.
Matrix Multiplication
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Now for the 2nd row of the first matrix
and the 1st column of the second matrix.
The result is placed in position a21.
Matrix Multiplication
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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.
Matrix Multiplication

So the result of multiplying our 2 matrices is as follows:
Matrix Multiplication

Multiply:
Answer

This is 2×3 times 3×2, which will give us a 2×2 answer
Multiply
Answer