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## Algebra (all content)

### Course: Algebra (all content)>Unit 20

Lesson 12: Introduction to matrix inverses

# Determining invertible matrices

Sal shows why a matrix is invertible if and only if its determinant is not 0. Created by Sal Khan.

## Want to join the conversation?

• in 8.30 are you sure its f/y and not f/d? • I see why a matrix is not invertible when its determinant is 0. However I don't understand the proof Sal used in the 2nd half of the video. For example, why does it prove that there is no matrix inverse when the 2 lines are parallel? • That's not correct. A solution is when two equations intersect at a point. If the lines were parallel and had different y-intercepts, the lines would never intersect and thus would have no solution.
If the parallel lines had the same y-intercepts, they would be on top of each other, representing an infinite number of solutions, as Sal says in the video. There wouldn't be a unique solution.
• I didn't know where to put the question, but will there be videos about 'square matrix diagonalisation'?
(sorry if the term isn't exact, i roughly translated it from the french term in my math lessons) • at how is it that you can compare the matrices
[a] and [b]
[c] ___ [d]
to the fractions a/c=b/d ??? I thought matrices were used to notate vectors, not ratios.... • So does it mean that if you were to use a 2x2 singular matrix as a transformation matrix, it will transform 2d space in a such a way, that both basis vectors will be pointing in the same direction? So it will kind of reduce the number of dimensions down to 1? • @~

What if e and f both equal zero? even if the determinant is zero, there could still be a solution at {x,y}=={0,0}.

Surely there's a name for that situation, though... • If e and f are both zero, there will be an infinite number of possible solutions.

|A| = 0 means that ad = bc or a/c = b/d. Select n = c/a, which gives c = n*a, then you get these equation
a/(n*a) = b/d
reduce and rearrange
d = n*b

The resulting equations become
a*x + b*y = 0
c*x + d*y = n*a*x + n*d*y = 0

Divide the second by n and you get these equations
a*x + b*y = 0
a*x + b*y = 0

They are the same, so for any x you can choose y = -a/b * x and both equations will hold. This actually holds for any f = n*e too (e and f both equal to zero is just a special case of this general principle). If f ≠ n*e, then there will be no solutions.

I hope this helps a bit.
• I dont understand a thing about a 2x2 matrix.
Well, suppose there's matrix [a,b c,d] so I understand that the first column would represent vector x and the second column would represent another y vector according to [x,y].
So this means we can write each vector as [a,b]*x+[c,d]*y. Those each of the vectors x and y can be plotted on the axes but how can we plot this whole term? x*some matrix+y*some matrix.
I hope you get my question.
(1 vote) • A matrix represents a transformation of the plane. If you have the matrix
a b
c d
that means that when you apply the matrix, the vector <1, 0> gets sent to <a, c> and the vector <0, 1> gets sent to <b, d>.

Then, once you know where those two vectors get sent, you can find out where any other vector <x, y> gets sent. It goes to x•<a,c>+y•<b, d>.   