To determine the area of the image of a rectangle under a given linear transformation, we need to understand how linear transformations affect areas. For a linear transformation represented by a matrix [tex]\( A \)[/tex], the area of the image of any shape, including a rectangle, is scaled by the absolute value of the determinant of [tex]\( A \)[/tex].
Given the transformation matrix:
[tex]\[
\begin{pmatrix}
-6 & 4 \\
7 & -8
\end{pmatrix}
\][/tex]
we will follow these steps to find the area of the image of the rectangle:
1. Calculate the determinant of the matrix:
For a 2x2 matrix:
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
the determinant is given by:
[tex]\[
\text{det}(A) = ad - bc
\][/tex]
2. Substitute the values from the matrix into the formula:
Here, [tex]\( a = -6 \)[/tex], [tex]\( b = 4 \)[/tex], [tex]\( c = 7 \)[/tex], and [tex]\( d = -8 \)[/tex].
[tex]\[
\text{det}(A) = (-6)(-8) - (4)(7)
\][/tex]
3. Carry out the multiplication and subtraction:
[tex]\[
\text{det}(A) = 48 - 28 = 20
\][/tex]
4. Take the absolute value of the determinant to find the scaling factor for the area:
The determinant we calculated is [tex]\( 20 \)[/tex], which is already positive. Thus the absolute value is:
[tex]\[
|\text{det}(A)| = 20
\][/tex]
Therefore, the area of the image of the rectangle under this transformation is [tex]\( 20 \)[/tex].
In summary, the original shape's area will be multiplied by the absolute value of the determinant of the transformation matrix. Since we determined the determinant to be 20, the area of the image of the rectangle is 20.
