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Once again, we remind the reader that not every system of equations can be solved by the matrix inverse method. Although the Gauss-Jordan method works for every situation, the matrix inverse method works only in cases where the inverse of the square matrix exists. In such cases the system has a unique solution.

We summarize our discussion in the following table.

    The method for finding the inverse of a matrix

  1. Write the augmented matrix A I n size 12{ left [ matrix { A {} # \lline {} # I rSub { size 8{n} } {}} right ]} {} .
  2. Write the augmented matrix in step 1 in reduced row echelon form.
  3. If the reduced row echelon form in 2 is I n B size 12{ left [ matrix { I rSub { size 8{n} } {} # \lline {} # B{}} right ]} {} , then B size 12{B} {} is the inverse of A size 12{A} {} .
  4. If the left side of the row reduced echelon is not an identity matrix, the inverse does not exist.

    The method for solving a system of equations when a unique solution exists

  1. Express the system in the matrix equation AX = B size 12{ ital "AX"=B} {} .
  2. To solve the equation AX = B size 12{ ital "AX"=B} {} , we multiply on both sides by A 1 size 12{A rSup { size 8{ - 1} } } {} .

Application of matrices in cryptography

In this section, we see a use of matrices in encoding and decoding secret messages. There are many techniques used, but we will use a method that first converts the secret message into a string of numbers by arbitrarily assigning a number to each letter of the message. Next we convert this string of numbers into a new set of numbers by multiplying the string by a square matrix of our choice that has an inverse. This new set of numbers represents the coded message. To decode the message, we take the string of coded numbers and multiply it by the inverse of the matrix to get the original string of numbers. Finally, by associating the numbers with their corresponding letters, we obtain the original message.

In this section, we will use the correspondence where the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27, and all punctuation is ignored.

A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13
N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26

Use the matrix A = 1 2 1 3 size 12{A= left [ matrix { 1 {} # 2 {} ##1 {} # 3{} } right ]} {} to encode the message: ATTACK NOW!

We divide the letters of the message into groups of two.

AT TA CK –N OW

We assign the numbers to these letters from the above table, and convert each pair of numbers into 2 × 1 size 12{2 times 1} {} matrices. In the case where a single letter is left over on the end, a space is added to make it into a pair.

A T = 1 20 size 12{ left [ matrix { A {} ##T } right ]= left [ matrix { 1 {} ##"20" } right ]} {} T A = 20 1 size 12{ left [ matrix { T {} ##A } right ]= left [ matrix { "20" {} ##1 } right ]} {} C K = 3 11 size 12{ left [ matrix { C {} ##K } right ]= left [ matrix { 3 {} ##"11" } right ]} {} _ N = 27 14 size 12{ left [ matrix { _ {} ##N } right ]= left [ matrix { "27" {} ##"14" } right ]} {} O W = 15 23 size 12{ left [ matrix { O {} ##W } right ]= left [ matrix { "15" {} ##"23" } right ]} {}

So at this stage, our message expressed as 2 × 1 size 12{2 times 1} {} matrices is as follows.

1 20 20 1 3 11 27 14 15 23 size 12{ left [ matrix { 1 {} ##"20" } right ]left [ matrix { "20" {} ##1 } right ]left [ matrix { 3 {} ##"11" } right ]left [ matrix { "27" {} ##"14" } right ]left [ matrix { "15" {} ##"23" } right ]} {}

Now to encode, we multiply, on the left, each matrix of our message by the matrix A. For example, the product of A with our first matrix is

1 2 1 3 1 20 = 41 61 size 12{ left [ matrix { 1 {} # 2 {} ##1 {} # 3{} } right ]left [ matrix { 1 {} ##"20" } right ]= left [ matrix { "41" {} ##"61" } right ]} {}

By multiplying each of the matrices in ( I ) by the matrix A, we get the desired coded message given below.

41 61 22 23 25 36 55 69 61 84 size 12{ left [ matrix { "41" {} ##"61" } right ]left [ matrix { "22" {} ##"23" } right ]left [ matrix { "25" {} ##"36" } right ]left [ matrix { "55" {} ##"69" } right ]left [ matrix { "61" {} ##"84" } right ]} {}
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Decode the following message that was encoded using matrix A = 1 2 1 3 size 12{A= left [ matrix { 1 {} # 2 {} ##1 {} # 3{} } right ]} {} .

21 26 37 53 45 54 74 101 53 69 size 12{ left [ matrix { "21" {} ##"26" } right ]left [ matrix { "37" {} ##"53" } right ]left [ matrix { "45" {} ##"54" } right ]left [ matrix { "74" {} ##"101" } right ]left [ matrix { "53" {} ##"69" } right ]} {}

Since this message was encoded by multiplying by the matrix A in [link] , we decode this message by first multiplying each matrix, on the left, by the inverse of matrix A given below.

A 1 = 3 2 1 1 size 12{A rSup { size 8{ - 1} } = left [ matrix { 3 {} # - 2 {} ##- 1 {} # 1{} } right ]} {}

For example,

3 2 1 1 21 26 = 11 5 size 12{ left [ matrix { 3 {} # - 2 {} ##- 1 {} # 1{} } right ]left [ matrix { "21" {} ##"26" } right ]= left [ matrix { "11" {} ##5 } right ]} {}

By multiplying each of the matrices in ( II ) by the matrix A 1 size 12{A rSup { size 8{ - 1} } } {} , we get the following.

11 5 5 16 27 9 20 27 21 16 size 12{ left [ matrix { "11" {} ##5 } right ]left [ matrix { 5 {} ##"16" } right ]left [ matrix { "27" {} ##9 } right ]left [ matrix { "20" {} ##"27" } right ]left [ matrix { "21" {} ##"16" } right ]} {}

Finally, by associating the numbers with their corresponding letters, we obtain the following.

K E E P _ I T _ U P size 12{ left [ matrix { K {} ##E } right ]left [ matrix { E {} ##P } right ]left [ matrix { _ {} ##I } right ]left [ matrix { T {} ##_ } right ]left [ matrix { U {} ##P } right ]} {}

And the message reads: KEEP IT UP.

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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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