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### 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 think it should be f/d. (From - )
• 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....
• I think he's just trying to get us to notice the correlation, not necessarily that it is in and of itself something to fret over, but it just makes it easier to identify the matrix at a glance before we do the math to figure it out.
(1 vote)
• 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?
• That's correct. A matrix is singular (noninvertible) because applying it to the 2D plane throws away some information, and we represent this by compressing the plane into a 1D line.
• @~

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>.