Let f be the linear transformation represented by the matrix
A=(-2 3
4 -4)
a=-2, b=3, c=4, d=-4
Find the point (x,y) such that f(x,y)=(4,4)
A*(x,y)T = (4,4)T
(x,y)T = A-1(4,4)T
To find the point (x, y) such that f(x, y) = (4, 4), we can use the equation f(x, y) = A * [x, y]^T, where A is the given matrix representing the linear transformation.
Let's denote the point (x, y) as [x, y]^T to emphasize that it is a column vector. Now we can rewrite the equation as A * [x, y]^T = [4, 4]^T.
Substituting the values of A and [x, y]^T into the equation, we get:
[ -2 3 ] [ x ] [ 4 ]
[ 4 -4 ] * [ y ] = [ 4 ]
Using matrix multiplication, we can calculate the equation as:
[ -2x + 3y ] [ 4 ]
[ 4x - 4y ] = [ 4 ]
Equating the corresponding elements, we get the following system of equations:
-2x + 3y = 4 ---- (1)
4x - 4y = 4 ---- (2)
Now, we can solve this system of equations to find the values of x and y.
From equation (1):
-2x + 3y = 4
Add 2x to both sides:
3y = 2x + 4
Divide both sides by 3:
y = (2/3)x + (4/3) ---- (3)
From equation (2):
4x - 4y = 4
Divide both sides by 4:
x - y = 1
Add y to both sides:
x = y + 1 ---- (4)
Now, we can substitute equation (4) into equation (3):
y = (2/3)(y + 1) + (4/3)
Multiply through by 3 to get rid of the fractions:
3y = 2(y + 1) + 4
Distribute 2 to the terms inside the parentheses:
3y = 2y + 2 + 4
Combine like terms:
3y = 2y + 6
Subtract 2y from both sides:
y = 6
Now, substitute the value of y back into equation (4) to find x:
x = 6 + 1
x = 7
Therefore, the point (x, y) such that f(x, y) = (4, 4) is (7, 6).