Inverse of a Square Matrix

Сентябрь 13, 2022

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Inverse of a Square Matrix

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2. Barnett/Ziegler/Byleen Finite Mathematics 11e Identity Matrix for Multiplication 1 is called the multiplicative identity for real
3. Barnett/Ziegler/Byleen Finite Mathematics 11e Identity Matrices 2 x 2 identity matrix: 3 x 3 identity matrix
4. Barnett/Ziegler/Byleen Finite Mathematics 11e Identity Matrix Multiplication AI = A (Verify the multiplication) We can also
5. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a Matrix All real numbers (excluding 0) have an inverse.
6. Barnett/Ziegler/Byleen Finite Mathematics 11e Matrix Inverses Some (not all) square matrices also have matrix inverses If
7. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a 2x2 Matrix There is a simple procedure to find
8. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a 2x2 matrix (continued) Step 1: Determine whether or not
9. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a 2x2 matrix (continued) Step 2. Reverse the entries on
10. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a 2x2 matrix (continued) The inverse of the matrix is
11. Barnett/Ziegler/Byleen Finite Mathematics 11e Inverse of a General Square Matrix 1. Augment the matrix with the
12. Barnett/Ziegler/Byleen Finite Mathematics 11e Example: Inverse of a 3x3 Matrix Find the inverse of Step 1.
13. Barnett/Ziegler/Byleen Finite Mathematics 11e Example (continued) Step 3. Multiply row 2 by (1/3) to get a
14. Barnett/Ziegler/Byleen Finite Mathematics 11e Example (continued) Step 7. Eliminate the (5/3) in the first row, third
15. Barnett/Ziegler/Byleen Finite Mathematics 11e Example Solution The inverse matrix appears on the right hand side of
16. Barnett/Ziegler/Byleen Finite Mathematics 11e Application: Cryptography Matrix inverses can provide a simple and effective procedure for
17. Barnett/Ziegler/Byleen Finite Mathematics 11e Any matrix A whose elements are positive integers and whose inverse exists
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Barnett/Ziegler/Byleen Finite Mathematics 11e
Identity Matrix for Multiplication
1 is called the

multiplicative identity for real numbers since a(1) = (1)a = a For example 5(1) = 5

A matrix is called square if it has the same number of rows and columns, that is, it has size n x n.
The set of all square matrices of size n x n also has a multiplicative identity In, with the property
AIn = InA = A
In is called the n x n identity matrix.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Identity Matrices
2 x 2 identity matrix:
3 x

3 identity matrix

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Identity Matrix Multiplication
AI = A (Verify the

multiplication)
We can also show that IA = A and in general AI = IA = A for all square matrices A.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a Matrix
All real numbers (excluding 0)

have an inverse.
For example

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Matrix Inverses
Some (not all) square matrices also have

matrix inverses
If the inverse of a matrix A exists, we shall call it A-1
Then

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a 2x2 Matrix
There is a

simple procedure to find the inverse of a two by two matrix. This procedure only works for the 2 x 2 case.
An example will be used to illustrate the procedure.
Example: Find the inverse of

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a 2x2 matrix (continued)
Step 1: Determine

whether or not the inverse actually exists. We define Δ = the difference of the product of the diagonal elements of the matrix.
In order for the inverse of a 2 x 2 matrix to exist, Δ cannot equal zero.
If Δ happens to be zero, then we conclude the inverse does not exist, and we stop all calculations.
In our case Δ = 2(2)-1(3) = 1, so we can proceed.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a 2x2 matrix (continued)
Step 2. Reverse the

entries on the main diagonal. In this example, both entries are 2, and no change is visible.
Step 3. Reverse the signs of the other diagonal entries 3 and 1 so they become -3 and -1
Step 4. Divide each element of the matrix by which in this case is 1, so no apparent change will be noticed.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a 2x2 matrix (continued)
The inverse of

the matrix is then
To verify that this is the inverse, we will multiply the original matrix by its inverse and hopefully obtain the 2 x 2 identity matrix:

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Inverse of a General Square Matrix
1. Augment the matrix with

the n x n identity matrix.
2. Use elementary row operations to transform the matrix on the left side of the vertical line to the n x n identity matrix. The row operations are used for the entire row, so that the matrix on the right hand side of the vertical line will also change.
3. When the matrix on the left is transformed to the n x n identity matrix, the matrix on the right of the vertical line is the inverse.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Example: Inverse of a 3x3 Matrix
Find the inverse

of
Step 1. Multiply R1 by (-2) and add the result to R2.
Step 2. Multiply R1 by 2 and add the result to R3

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
Step 3. Multiply row 2 by (1/3) to

get a 1 in the second row, first position.
Step 4. Add R2 to R1.
Step 5. Multiply R2 by 4 and add the result to R3.
Step 6. Multiply R3 by 3/5 to get a 1 in the third row, third position.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Example (continued)
Step 7. Eliminate the (5/3) in the first

row, third position by multiplying R3 by (-5/3) and adding result to R1.
Step 8. Eliminate the (-4/3) in the second row, third position by multiplying R3 by (4/3) and adding result to R2.
Step 9. You now have the identity matrix on the left, which is our goal.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Example Solution
The inverse matrix appears on the right

hand side of the vertical line and is displayed below. Many calculators as well as computers have software programs that can calculate the inverse of a matrix quite easily. If you have access to a TI 83, consult the manual to determine how to find the inverse using a calculator.

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Application: Cryptography
Matrix inverses can provide a simple and

effective procedure for encoding and decoding messages. To begin, assign the numbers 1-26 to the letters in the alphabet, as shown below. Also assign the number 0 to a blank to provide for space between words.
Blank A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Thus the message “SECRET CODE” corresponds to the sequence
19 5 3 18 5 20 0 3 15 4 5

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Barnett/Ziegler/Byleen Finite Mathematics 11e
Any matrix A whose elements are positive integers

and whose inverse exists can be used as an encoding matrix. For example, to use the 2 x 2 matrix
to encode the message above, first divide the numbers in the sequence 19 5 3 18 5 20 0 3 15 4 5 into groups of 2, and use these groups as the columns of a matrix B:

Cryptography (continued)

We added an extra blank at the end of the message to make the columns come out even.

Inverse of a Square Matrix

Содержание

Barnett/Ziegler/Byleen Finite Mathematics 11eIdentity Matrix for Multiplication 1 is called the

Barnett/Ziegler/Byleen Finite Mathematics 11eIdentity Matrices2 x 2 identity matrix: 3 x

Barnett/Ziegler/Byleen Finite Mathematics 11eIdentity Matrix Multiplication AI = A (Verify the

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a MatrixAll real numbers (excluding 0)

Barnett/Ziegler/Byleen Finite Mathematics 11eMatrix InversesSome (not all) square matrices also have

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a 2x2 Matrix There is a

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a 2x2 matrix (continued) Step 1: Determine

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a 2x2 matrix (continued)Step 2. Reverse the

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a 2x2 matrix (continued) The inverse of

Barnett/Ziegler/Byleen Finite Mathematics 11eInverse of a General Square Matrix1. Augment the matrix with

Barnett/Ziegler/Byleen Finite Mathematics 11eExample: Inverse of a 3x3 Matrix Find the inverse

Barnett/Ziegler/Byleen Finite Mathematics 11eExample (continued)Step 3. Multiply row 2 by (1/3) to

Barnett/Ziegler/Byleen Finite Mathematics 11eExample (continued)Step 7. Eliminate the (5/3) in the first

Barnett/Ziegler/Byleen Finite Mathematics 11eExample Solution The inverse matrix appears on the right

Barnett/Ziegler/Byleen Finite Mathematics 11eApplication: CryptographyMatrix inverses can provide a simple and

Barnett/Ziegler/Byleen Finite Mathematics 11eAny matrix A whose elements are positive integers

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