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## Linear algebra

### Course: Linear algebra>Unit 2

Lesson 5: Finding inverses and determinants

# Deriving a method for determining inverses

Determining a method for constructing inverse transformation matrices. Created by Sal Khan.

## Want to join the conversation?

• What is the difference between a transformation matrix and a permutation matrix?
• A permutation matrix has ones and zeroes only. All it can do is move entries from the matrix/vector it is being multiplied with, so some very limited transformations could be represented with a permutation vector. For instance, here's a permutation matrix to swap row 2 and row 1 in a matrix/vector with 3 rows:

[0 1 0]
[1 0 0]
[0 0 1]

it is the identity matrix with row 2 and row 1 swapped. Simple huh? Please multiply that matrix with this column vector

[x]
[y]
[z]

to verify that it does what we think it should do.

A transformation matrix has a lot more freedom. It can stretch along one axis independently from others. For instance, here's a stretch of 5 in the x axis and a shrink by a half in the z axis:

[ 5 0 0]
[ 0 1 0]
[ 0 0 1/2]

transformation matrices can rotate, flip, project, etc: http://en.wikipedia.org/wiki/Linear_mapping#Examples_of_linear_transformation_matrices.
• In my current textbook (and I'm sure other places discussing this topic),
invertible is a term that means the same thing as the term non-singular, such that there are a finite number of row operations you can do to get to the identity matrix. (All those row operations merged together are called the inverse of the matrix.)
Likewise, non-invertible corresponds with singular, such that there is no matrix that can produce the identity matrix for a singular matrix.
It's been pretty confusing for me with different terminology in my class and online, but I hope this helps out someone!
• Mathematics progresses at different rates among different people with different perspectives for different reasons all the time, so you end up with differences in language, interpretation, visualization, focus, goal and strategy... all of which leads to differences in vocabulary. You will (hopefully) find that each type of vocabulary reveals the math in a different way, so it is useful to master as many versions of an idea as you can.
• I do not understand how he got the numbers at 3min 38 seconds Can you plz explain
• S is the transformation matrix we're trying to solve, I is the Identity matrix,
the idea is that if we apply the transformation to I, that is, SxI, we should get S itself since I is identity.
So let us apply to I (identity) what we know of what to do regarding this transformation, [a1, a2, a3] -> [a1, a2+a1, a3-a1], the result should be S.
• Can anyone please help me out with a way to calculate faster the inverse of a matrix?
• I'm having a problem. I've multiplied S1xA by hand and with an app. Both times I got
1 -1 -1
-2 2 3
-2 1 4

Not
1 -1 -1
0 1 2
0 2 5

Any thoughts on what I did wrong?
• Wrong order. Unlike normal multiplication A*B is not the same as B*A

If you are confused which order you should do, think of each matrix as a function, where you start with f(x) and then to have that be part of a function you write g(f(x)), expanding to the left. That's how I think of it.
• What is row echelon form?
(1 vote)
• Hat is homomorphism? (This is out of context question)
• A homomorphism is a map between two algebraic structures of the same type (that can be vector spaces), preserving the structures' operations. This means a function f: A -> B mapping from vector space A to B, f is a homomorphism if f(x.y) = f(x).f(y) for every x,y of A.
You can say that f is a homomorphism if f preserves operations.

For example: We have a 2x2 matrix A. A =
[a 0
0 a]

f: A -> B

f(a+b) =
[a+b 0
0 a+b]

= f(a) + f(b)

and

f(2a) =
[2a 0
0 2a]

= 2f(a)

->> f is a homomorphism because f preserves matrix addition and multiplication.
(1 vote)
• Is it possible to convert a matrix into row echelon form by transforming the row vectors instead of column vectors?