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Is the linear transformation matrix T invertible

  • Thread starter war485
  • Start date
92
0
1. 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.

2. Homework Equations

none

3. 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?
 
316
0
What is the inverse of T? Hint: T^-1 will look a lot like T, but with something replaced by its inverse.
 
32,580
4,310
1. 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.

2. Homework Equations

none

3. 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.
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?
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.


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.
 
92
0
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 ]
?
 
32,580
4,310
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).
 
92
0
It's starting to look really aweful. Is there another way to find the inverse?
 

Dick

Science Advisor
Homework Helper
26,258
618
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)?
 
92
0
what's that X in S(X)=XB^(-1) ?
 

Dick

Science Advisor
Homework Helper
26,258
618
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.
 
92
0
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.
 

Dick

Science Advisor
Homework Helper
26,258
618
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.
 

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