18.06 Linear Algebra, Spring 2010
Transcript – Lecture 20
OK, this is lecture twenty. And this is the final lecture on determinants. And it's
about the applications. So we worked hard in the last two lectures to get a formula
for the determinant and the properties of the determinant. Now to use the
determinant and, and always this determinant packs all this information into a single
number.
And that number can give us formulas for all sorts of, things that we've been
calculating already without formulas. Now what was A inverse? So, so I'm beginning
with the formula for A inverse. Two, two by two formula we know, right? The two by
two formula for A inverse, the inverse of a b c d inverse is one over the determinant
times d a -b -c. Somehow I want to see what's going on for three by three and n by
n.
And actually maybe you can see what's going on from this two by two case. So
there's a formula for the inverse, and what did I divide by? The determinant. So
what I'm looking for is a formula where it has one over the determinant and, and
you remember why that makes good sense, because then that's perfect as long as
the determinant isn't zero, and that's exactly when there is an inverse. But now I
have to ask can we recognize any of this stuff? Do you recognize what that number d
is from the past? From last, from the last lecture? My hint is think cofactors.
Because my formula is going to be, my formula for the inverse is going to be one
over the determinant times a matrix of cofactors. So you remember that D? What's
that the cofactor of? Remember cofactors? If -- that's the one one cofactor, because
if I strike out row and column one, I'm left with d.
And what's minus b? OK.
Which cofactor is that one? Oh, minus b is the cofactor of c, right? If I strike out the
c, I'm left with a b there.
And why the minus sign? Because this c was in a two one position, and two plus one
is odd.
So there was a minus went into the cofactor, and that's it. OK.
I'll write down next what my formula is.
Here's the big formula for the A -- for A inverse.
It's one over the determinant of A and then some matrix.
And that matrix is the matrix of cofactors, c. Only one thing, it turns -- you'll see, I
have to, I transpose.