Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes [1 2 15 18 20 0 13 9 19 19 9 15 14]. To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix 3 4 2 1 . We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A. Encrypted Matrix = 3 4 2 1 1 15 20 13 19 9 14 2 18 0 9 19 15 0 = 11 117 60 75 133 87 42 4 48 40 35 57 33 28 , which we can also write as [11 4 117 48 60 40 75 35 133 57 87 33 42 28]. To decipher the encoded message, multiply the encrypted matrix by A−1. The following question uses the above matrix A for encoding and decoding.Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.) [69 21 126 54 27 13 60 40 59 16 149 61 87 28]
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take < space > = 0, A = 1, B = 2, and so on. Thus, for example, "ABORT MISSION" becomes
To encrypt this coded phrase, we use an invertible matrix of any size with integer entries. For instance, let us take A to be the 2 × 2 matrix
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3 | 4 |
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| 2 | 1 |
We can first arrange the coded sequence of numbers in the form of a matrix with two rows (using zero in the last place if we have an odd number of characters) and then multiply on the left by A.
| Encrypted Matrix | = |
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which we can also write as
To decipher the encoded message, multiply the encrypted matrix by
The following question uses the above matrix A for encoding and decoding.
Decode the following message, which was encrypted using the matrix A. (Include any appropriate spaces in your answer.)
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