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Inverse of a MatrixPlease read our Introduction to Matrices first. What is the Inverse of a Matrix? The Inverse of a Matrix is the same idea as the reciprocal of a number:
But we don't write 1/A (because we don't divide by a Matrix!), instead we write A-1 for the inverse: (In fact 1/8 can also be written as 8-1) And there are other similarities: When you multiply a number by its reciprocal you get 1 8 × (1/8) = 1 When you multiply a Matrix by its Inverse you get theIdentity Matrix (which is like "1" for Matrices): A × A-1 = I It also works when the inverse comes first: (1/8) × 8 = 1 and A-1 × A = I Identity Matrix Note: the "Identity Matrix" is the matrix equivalent of the number "1":
The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc ... Definition So we have a definition of a Matrix Inverse ... The Inverse of A is A-1 only when: A × A-1 = A-1 × A = I Sometimes there is no Inverse at all. X2 Matrix OK, how do we calculate the Inverse? Well, for a 2x2 Matrix the Inverse is: In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc). Let us try an example: How do we know this is the right answer? Remember it must be true that: A × A-1 = I So, let us check to see what happens when we multiply the matrix by its inverse: And, hey!, we end up with the Identity Matrix! So it must be right. It should also be true that: A-1 × A = I Why don't you have a go at multiplying these? See if you also get the Identity Matrix:
Why Would We Want an Inverse? Because with Matrices we don't divide! Seriously, there is no concept of dividing by a Matrix. But we can multiply by an Inverse, which achieves the same thing. Imagine you couldn't divide by numbers, and someone asked "How do I share 10 apples with 2 people?" But you could take the reciprocal of 2 (which is 0.5), so you could answer: 10 × 0.5 = 5 They get 5 apples each The same thing can be done with Matrices: Say that you know Matrix A and B, and want to find Matrix X: XA = B It would be nice to divide both sides by A (to get X=B/A), but remember we can't divide.
But what if we multiply both sides by A-1 ? XAA-1 = BA-1 And we know that AA-1 = I, so: XI = BA-1 We can remove I (for the same reason we could remove "1" from 1x = ab for numbers): X = BA-1 And we have our answer (assuming we can calculate A-1) In that example we were very careful to get the multiplications correct, because with Matrices the order of multiplication matters. AB is almost never equal to BA. Date: 2015-12-11; view: 1391
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