Find the matrix M of the inverse linear transformation, T^-1?

Question

Im trying to solve this question. but im stuck and cant figure it out. Can someone help please
T:R^2 -> R^2

T(X)=
[ 0 -4] X
[-1 -4]

Find the matrix M of the inverse linear transformation, T^-1

M= [ _ _ ]
[ _ _ ]

Answer 1

One of the nice things about representing a linear transformation T by its standard matrix A is that T has an inverse if and only if A has an inverse. So we calculate A^-1, and we do get an answer:
[...1...-1]
[-1/4...0]
(periods for spacing).

So T^-1(x) = A^-1 * X, or M=A^-1

Answer 2

First, find the determinant of the matrix. In your case, it's equal to -4 (if you don't know how to find a determinant by now, you is in *big* trouble)

Next, form the matrix of cofactors by replacign each element of the matrix with the determinant of its minor multiplied by the sign of the permutation (or the 'signum' function, (-1)^(i+j)) and, again, if you don't know those words, or what they mean, you *really* need to get busy studying.

In your problem, it would be
|-4.....1|
|.4.....0|

Now, form the transpose of this to get
|-4.....4|
|.1.....0|
and divide each element by the determinant of the original matrix
|.1......-1|
|-1/4....0|
This is the inverse of the original matrix. You can check by multiplying it by the original matrix to get the identity matrix
|1...0|
|0...1|

And the steps I outlined above work with square matrices of *any* order.

Remember that any n X n matrix is a linear transform taking a vector in n-space into another vector in n-space unless the matrix is singular (which means its determinant will be 0)

HTH ☺


Doug

Answer 3

multiply by -0.25 the following matrix



-4 4
1 0

notice the pattern, and that the number -0.25 is the reciprocal of the determinate of the given matrix.