If M=(abcd) is a 2-by-2 matrix with ad-bc≠0 then the matrix M-1=1ad-bc(d-b-ca) satisfies M-1M=MM-1=I . We say that M-1 is the inverse of M .
20. Inverses
20. Inverses
In the next few videos, we're going to answer the question: can you divide by a matrix?
This is the analogue of the reciprocal x-1=1/x of a number: the equation M-1M=I is the analogue of x-1x=1 .
Let's just check M-1M=I (check the other equality for yourself). 1ad-bc(d-b-ca)(abcd)=1ad-bc(da-bcdb-bd-ca+ac-cb+ad)=(1001)=I.
We can use this to "divide" by a matrix: if we have a matrix equation AB=C then we can multiply both sides on the left by A-1 to get A-1AB=A-1C , and since A-1A=I this means B=A-1C .
With great power comes great responsibility: you should never write BA for matrices A,B ! It's not clear if you're doing A-1B or BA-1 (and these are different because A-1 and B might not commute).
We can use inverses to solve simultaneous equations. For example: x-y=-1,x+y=3
Bigger matrices
We'd like to generalise the notion of inverse to bigger matrices.
Let A be an n -by-n matrix. We say that A is invertible if there exists a matrix B such that AB=BA=I . (Here I is the n -by-n identity matrix). If such a B exists, then it's unique, so we're justified in calling it the inverse of A and writing it as A-1 .
To see that the inverse is unique when it exists, suppose you had two inverses B,C for A : AB=AC=I,BA=CA=I.
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BAC=(BA)C=IC=C,
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BAC=B(AC)=BI=B,
so C=BAC=B .
If A,B are invertible matrices then AB is invertible with inverse (AB)-1=B-1A-1 .
we have (AB)(B-1A-1)=A(BB-1)A-1=AIA-1=AA-1=I
Had we tried to use A-1B-1 instead, we would have obtained ABA-1B-1 , and we couldn't have cancelled anything because the various terms don't commute.
We saw for 2-by-2 matrices that a matrix is invertible if ad-bc≠0 . Even more basically, a 1-by-1 matrix (x) is invertible if and only if x≠0 . We'll see in a few videos' time that there is an analogous quantity (the determinant) det(A) which is nonzero precisely when A is invertible. First though, we're going to explain how to calculate inverses of n -by-n matrices.