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Inversion: A Caution, Warning: Not all matrices can be inverted. Recall that the inverse of a regular number is its reciprocal, so 4/3 is the inverse of 3/4, 2 is the inverse of 1/2, and so forth. But there is no inverse for 0, because you cannot flip 0/1 to get 1/0 (since division by zero doesn't work). For similar reasons (which you may or may not encounter in later studies), some matrices cannot be inverted. Given a matrix A, the inverse A^{–1} (if said inverse matrix in fact exists) can be multiplied on either side of A to get the identity. That is, AA^{–1} = A^{–1}A = I. Keeping in mind the rules for matrix multiplication, this says that A must have the same number of rows and columns; that is, A must be square. (Otherwise, the multiplication wouldn't work.) If the matrix isn't square, it cannot have a (properly twosided) inverse. However, while all invertible matrices are square, not all square matrices are invertible.
Always be careful of the order in which you multiply matrices. For instance, if you are given B and C and asked to solve the matrix equation AB = C for A, you would need to cancel off B. To do this, you would have to multiply B^{–1} on B; that is, you would have to multiply on the right: AB = C
The side on which you multiply
will depend upon the exercise. Take the time to get this right. There is only one "word problem" sort of exercise that I can think of that uses matrices and their inverses, and it involves coding and decoding.
Translate the coded message:
To do the decoding, I have to undo the matrix multiplication. To undo the multiplication, I need to multiply by the inverse of the encoding matrix. So my first step is to invert the coding matrix: So the inverse matrix is: My correspondent converted letters to numbers, and then entered those numbers into a matrix C. He then multiplied by this matrix by the encoding matrix A, and sent me the message matrix M. Since the encoding was done by multiplying C on the left, then I know the encoding equation was: AC = M To reverse the encoding, I need to multiply by A^{–1} on the left: A^{–1}AC
= A^{–1}M
This gives me: At this point, the solution is a simple matter of doing the numbertoletter correspondence:
T H E L A W I S A N A ... (You can complete the decoding to view the original quotation.) A better "code" could be constructed by shifting the letters first, adding some value to each letter's coded result, using a larger invertible matrix, etc, etc. The above example is fairly simplistic, and is intended only to show you the general methodology. << Previous Top  1  2  Return to Index



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