For a square matrix 3 x 3, determining whether it is invertible or not will involve a system of linear equations with 9 equations and 9 variables. Therefore solving this for the unknowns becomes quite complicated and tedious. Instead, the shortcut method exists whereby M|I is turned into I|B and if this exists, then B is automatically M-1 (inverse of matrix M). I is the identity matrix.
Application: Being able to solve for the inverse of a matrix allows for simple solutions to many practical problems.
I. Cryptography
If we assign numbers 1 to 26 for the letters of the alphabet, and 0 to blank (space), then we can encode and decode any message by doing the following:
1. Create a matrix of choice M (encoding matrix) with m x n (i.e. 3 x 3)
2. Assign a number to each letter in the encoded message. Then, create a matrix 3 x n where n can be any number of columns. Call this matrix C.
3. Multiply M by C. This is the encoded message.
4. To decode this message, one must find matrix B (M-1, inverse of M) and multiply B by C to create matrix D, in that order since matrices are not commutative (order matters in multiplication).
5. Assigning letters to the elements of the resulting matrix, D, going down in the columns sequentially, will lead to the decoded message.
It is said that while this may seem obvious and trivial, actually decoding messages encoded in this way is not that easy!
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