Is the linear transformation matrix T invertible

T(A)*S(A) is the identity matrix, but you have to multiply it the right way.I'm not sure what you're doing there, but it doesn't look right. Start with S(X)=XB^(-1) and write out S(T(A)) in terms of A.[S(T(A))] = [S(AB)] = [AB][B^(-1)] = [A][BB^(-1)] = [A][I] = ATherefore, S(T(A)) = A, and S = T^(-1).In summary, the linear transformation matrix T is invertible with respect to the standard bases and its inverse is given by S(X)=XB^(-1).
  • #1
war485
92
0

Homework Statement



T: M22 --> M22 defined by T(A) = AB where B =
[ 3 2 ]
[ 2 1 ]

Is the linear transformation matrix T invertible with respect to the standard bases? If so, find it.

Homework Equations



none

The Attempt at a Solution



This is going to sound stupid, but I need help in finding what the transformation matrix T actually is before I can say if it is invertible or not. So far I got:

T(A) = AB =
[a b ] [ 3 2 ]
[c d ] [ 2 1 ]

=
[ 3a+2b 2a+b ]
[ 3c+2d 2c+d ]

and I found T(e11) =
[3 2 ]
[0 0 ]

T(e12) =
[ 2 1 ]
[ 0 0 ]

T(e21) =
[ 0 0 ]
[ 3 2 ]

T(e22) =
[ 0 0 ]
[ 2 1 ]

and now I'm stuck. Thinking ahead a little, I know that if the kernal of T = 0, it will be invertible, or if the determinant of the matrix is not 0, it is invertible.

P.S. is there an easier way to write matrices here?
 
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  • #2
What is the inverse of T? Hint: T^-1 will look a lot like T, but with something replaced by its inverse.
 
  • #3
war485 said:

Homework Statement



T: M22 --> M22 defined by T(A) = AB where B =
[ 3 2 ]
[ 2 1 ]

Is the linear transformation matrix T invertible with respect to the standard bases? If so, find it.

Homework Equations



none

The Attempt at a Solution



This is going to sound stupid, but I need help in finding what the transformation matrix T actually is before I can say if it is invertible or not.
No, it doesn't sound stupid to me. This transformation takes a 2x2 matrix A as input, and produces another 2x2 matrix AB as its output, where B is as you showed.
war485 said:
So far I got:

T(A) = AB =
[a b ] [ 3 2 ]
[c d ] [ 2 1 ]

=
[ 3a+2b 2a+b ]
[ 3c+2d 2c+d ]

and I found T(e11) =
[3 2 ]
[0 0 ]

T(e12) =
[ 2 1 ]
[ 0 0 ]

T(e21) =
[ 0 0 ]
[ 3 2 ]

T(e22) =
[ 0 0 ]
[ 2 1 ]

and now I'm stuck.
I think what is meant by asking if T is invertible with respect to the standard basis is this: Your basis vectors (the matrices eij) are linearly independent. Are the new vectors (the matrices T(eij)) also linearly independent?
war485 said:
Thinking ahead a little, I know that if the kernal of T = 0, it will be invertible, or if the determinant of the matrix is not 0, it is invertible.
The determinant of AB is zero some of the time, and this is completely determined by the matrix A. This means that whether T is invertible depends completely on the matrix A that is used. Calculate the determinant of AB to see this.


war485 said:
P.S. is there an easier way to write matrices here?

There's a nicer way using tex script, but I wouldn't call it easier.
 
  • #4
Mark44 said:
I think what is meant by asking if T is invertible with respect to the standard basis is this: Your basis vectors (the matrices eij) are linearly independent. Are the new vectors (the matrices T(eij)) also linearly independent?

Yes they're also linearly independent.

Is the inverse =
[ -1 2 ] [ a b ]
[ 2 -3 ] [ c d ]

=
[ 2c-a 2d-b ]
[ 2a-3c 2b-3d ]
?
 
  • #5
It's easy enough to check. Multiply what you got by AB, and it should come out to I.

I think I remember a theorem that says, if A and B are invertible, then (AB)^(-1) = B^(-1)A^(-1). The matrix you show on the left in your product looks like it might be B^(-1), but the one on the right definitely isn't A^(-1).
 
  • #6
It's starting to look really aweful. Is there another way to find the inverse?
 
  • #7
You don't actually need to find the matrix corresponding to T. You just have to define it's inverse. It looks like you've already found the inverse of B. Think about S(X)=XB^(-1). Could S be T^(-1)?
 
  • #8
what's that X in S(X)=XB^(-1) ?
 
  • #9
X is any matrix, just like the 'A' in T(A)=AB. I could have written S(A)=AB^(-1) as well, they both mean the same thing.
 
  • #10
This is what I got after what you said:
[ a b ] [ -1 2 ]
[ c d ] [ 2 -3 ]
=
-a+2b 2a-3b
-c+2d 2c-3d
You hinted towards this may be being the inverse and I multiplied it out by

[ 3a+2b 2a+b ]
[ 3c+2d 2c+d ]

And it doesn't look like the inverse in the end.
 
  • #11
S is the inverse of T if S(T(A))=A=T(S(A)). S(T(A))=S(AB)=ABB^(-1)=A. You seem to be multiplying T(A)*S(A) and trying to get A or something. That's not what the inverse means.
 

1. What does it mean for a linear transformation matrix to be invertible?

For a linear transformation matrix T to be invertible, it means that there exists another matrix, called the inverse of T, that when multiplied with T yields the identity matrix. In other words, the inverse of T "undoes" the transformation performed by T.

2. How can I determine if a linear transformation matrix T is invertible?

A linear transformation matrix T is invertible if and only if its determinant is non-zero. This means that the columns of T are linearly independent, and the transformation performed by T is one-to-one and onto.

3. What happens if a linear transformation matrix T is not invertible?

If a linear transformation matrix T is not invertible, it means that the columns of T are linearly dependent, and the transformation performed by T is not one-to-one. This can lead to loss of information and difficulties in solving equations involving T.

4. Can a linear transformation matrix T be invertible if it has non-square dimensions?

No, a linear transformation matrix T can only be invertible if it is a square matrix (same number of rows and columns). This is because the inverse of T must have the same dimensions as T, and non-square matrices do not have an inverse.

5. Are there any practical applications for invertible linear transformation matrices?

Yes, invertible linear transformation matrices are commonly used in various fields such as computer graphics, engineering, and economics. They can be used to perform rotations, scaling, shearing, and other transformations on objects and data, and their invertibility ensures that the original information can be recovered.

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