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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 rowcolumn elements from each Matrix to produce a new
element
in the same rowcolumn 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 rowcolumn 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 rowcolumn elements from each Matrix to produce a new
element
in the same rowcolumn 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 rowcolumn 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 rowcolumn location identified.
5. Continue until all rowcolumn 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 1column 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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