An object of area 20 cm²undergoes a transformation given by the matrix (-1 -2)
(4 3)
followed by (2 3) find the area of the final
(-1 2).
image
To find the area of the final image, we need to calculate the determinant of the resulting matrix.
The given matrix is:
A = (-1 -2)
(4 3)
Multiplying A by the given transformation matrix:
B = (2 3)
(-1 2)
We have:
C = AB
= (-1 -2)(2 -1) + (-2 3)(3 2)
= (0 -8) + (3 6)
= (3 -2)
The determinant of C is:
det(C) = 3 * 2 - (-2) * (-3)
= 6 - 6
= 0
The area of the final image is 0 cm².
To find the area of the final image after the given transformations, we first need to find the matrix resulting from the composition of the two matrices.
Let the first matrix be A = (-1 -2, 4 3) and the second matrix be B = (2 3, -1 2).
To find the resulting matrix, we need to multiply the two matrices:
AB = (-1 -2, 4 3) * (2 3, -1 2)
Using matrix multiplication, we can calculate:
AB = (-1*(-1) + -2*(-1), -1*3 + -2*2), (4*(-1) + 3*(-1), 4*3 + 3*2)
Simplifying further, we get:
AB = (1 -7, -7 18)
Therefore, the resulting matrix after the two given transformations is:
AB = (1 -7, -7 18)
Now, to find the area of the final image, we need to calculate the determinant of the resulting matrix. The determinant of a 2x2 matrix is given by |A| = ad - bc.
In this case, where A = (1 -7, -7 18), we have:
|A| = (1*18) - (-7*-7)
= 18 - 49
= -31
Since the determinant is negative, the resulting image is a reflection of the original object. The absolute value of the determinant gives us the scale factor of the area:
Area of final image = |A| × Area of original object
= |-31| × 20 cm²
= 31 × 20 cm²
= 620 cm²
Therefore, the area of the final image is 620 cm².