# MATRIX OPERATIONS

** Summary of article :
What is an operation?
Addition of two matrices .
Multiplication of a Matrix by a scalar.
Subtraction of two matrices : two ways to do it.
Combinations of Addition, Subtraction, Scalar Multiplication.
Matrix Multiplication**

An

**operation**is a way that we combine two elements.

The

**basic operations**are:

Addition (+)

Subtraction (-)

Multiplication ( * )

Division ( ÷ )

Here are some examples of combining elements and the operations used :

Combining elements that are rational numbers ( fractions ) using the operation of addition.

Combining elements that are second degree binomials using the operation of

multiplication.

Combining elements that are integers using the operation of division.

−35 ÷ 7

This FAQ will review the definitions of addition, scalar multiplication, subtraction, and

the multiplication of matrices. Division is undefined for matrices, but there is a separate

related concept, Inverse Matrices, that is similar to division and is found among the other

FAQ topics listed on the Online Math Center .

**ADDITION of TWO MATRICES
**

To add two matrices, their orders (the number of rows and columns in both matrices)

must be the same.

Add the corresponding row-column elements from each Matrix to produce a new element

in the same row-column location.

Example 1: Both matrices are order 2 x 2

Example 2: Both matrices are order 3 x 5

**MULTIPLICATION of a MATRIX by a scalar**

A scalar is simply a number . To multiply a Matrix by a scalar, distribute the
scalar to all

elements in the Matrix and multiply.

Example 1: Multiply Matrix A by the scalar 3:

Example 2: Multiply Matrix B by the scalar :

**SUBTRACTION of TWO MATRICES**

To subtract two matrices, their orders (the number of rows and columns in both
matrices)

must be the same.

**The easy way…**

To subtract Matrix A and B, simply subtract corresponding row-column elements.

Don’t forget to change signs of elements in Matrix B

Example 1: Subtract Matrix B from Matrix A. Both are order 2 x 2.

Example 2: Subtract Matrix B from Matrix A. Both are order 1 x 4.

Example 3: Subtract D from C

Matrix C is a 3 x 3 order Matrix. Matrix D is a 3 x 1
order Matrix.

Since the order of the two matrices is different , they can NOT be subtracted.

**Now the mathematics behind the scenes…**

The negative sign in front of the second Matrix is actually the scalar “ −1”.

Distribute the scalar “ −1” to all elements in the second Matrix.

Add the corresponding row-column elements from each Matrix to produce a new
element

in the same row-column location.

Example: Subtract B from A

**COMBINATIONS of ADDITION, SUBTRACTION, and Scalar
MULTIPLICATION.**

The order of operations requires multiplication be done before addition or subtraction, so

first multiply the elements inside a Matrix by the scalar in front of it. Add and/or subtract

afterwards.

Example:

Find 3A− 5B

**MULTIPLICATION of TWO MATRICES**

Two matrices A and B can be multiplied if the **number of columns in A **is
the** same** as

**the number of rows in B**. The new Matrix will have the same number of rows
as A and

the same number of columns as B.

**Example, find A* B**

Matrix A is 2 x **3 **(2 rows, **3 columns**).
Matrix B is a** 3** x 3 (**3 rows**, 3 columns) Matrix.

The columns in A equal the rows in B, so we can multiply A*B producing a 2 x 3
Matrix.

However, the columns in B do not equal the rows in A, so we can not multiply
B*A.

Steps in multiplying two matrices

1. Determine if the two matrices can be multiplied, i.e. the number of columns
in A

equals the number of rows in B.

If the order of A is (m x n) and the order of B is (n x p), the new Matrix will
be of

order (m x p).

2. Set up the new, blank (m x p) Matrix.

3. Pick a row-column location of an element in the new Matrix, e.g. the element
in

row 1 column 1 of the new Matrix.

4. Multiply the first element from the identified row in A by the first element
in the

identified column of B. Multiply the second element from the identified row of A

by the second element in the identified column of B. Continue across the row of
A

and down the column of B. Then add all of the results. Place the answer in the

new Matrix at the row-column location identified.

5. Continue until all row-column locations of the new Matrix are filled.

Example:

Matrix A has 2 rows and 2 columns; Matrix B has 2 rows and
2 columns. The number of

columns in A equals the number of rows in B, so the two matrices can be
multiplied.

The (2 x 2) times (2 x 2) will produce a new (2 x 2) Matrix. Set up the new,
blank 2 x 2

Matrix.

The question mark has been placed in the first row, first
column location of the new

Matrix. So multiply the first element in row 1 of Matrix A by the first element
in column

1 of Matrix B. Then multiply the second element in row 1 of Matrix A by the
second

element in column 1 of Matrix B.

Multplying row 1 of A by column 1 of B:

We have gone across row 1 in Matrix A and down column 1 in
Matrix B, so we can add

the results and place the answer in row 1-column 1 of the new Matrix

New Matrix: row 1, column 1:

Now let’s find the element for row 1, column 2:

Multplying row 1 of A by column 2 of B:

So row 1, column 2 of the new Matrix is the element 33:

For **row 2, column 1** multiply row 2 of Matrix A and
column 1 of Matrix B.

For** row 2, column 2** multiply row 2 of Matrix A and column 2 of matrix B:

Hence:

**Example: Find A * B **

A is order (1 x 3) and B is order (3 x 1), so the new
matrix will be order (1 x 1): [?]

There is only one element in the new matrix: the row 1, column 1 element.

Multiply row 1 elements of Matrix A by column 1 elements of Matrix B and add the

result.

**Example: Find A * B**

A is order (2 x 3) and B is order (3 x 2), so the new matrix will be order (2 x 2):

Row 1, Column 1: (1)(5) + (0)(1) + (4)(7) = 5 + 0 + 28 =
33

Row 1, Column 2: (1)(0) + (0)(6) + (4)(−2) = 0 + 0 −8 = −8

Row 2, Column 1: (−1)(5) + (2)(1) + (3)(7) = −5 + 2 + 21 =18

Row 2, Column 2: (−1)(0) + (2)(6) + (3)(−2) = 0 +12 − 6 = 6

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