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To find the area of the transformed region, multiply the original area by the absolute value of the determinant of the transformation matrix. The formula is: $\text{New Area} = \text{Old Area} \cdot |\det(A)|$.

Conditions: The transformation is linear and represented by $a 2\times 2$ matrix.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.

To find the area of the transformed region, multiply the original area by the absolute value of the determinant of the transformation matrix. The formula is: $\text{New Area} = \text{Old Area} \cdot |\det(A)|$.

Conditions: The transformation is linear and represented by $a 2\times 2$ matrix.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.

The determinant scales the area of any measurable planar figure, not just squares. The absolute value of the determinant acts as a uniform area scaling factor for all regions under the linear transformation.

Conditions: The transformation is linear.; The figure is measurable and has finite area.; Use ordinary Euclidean area in standard orthonormal coordinates.

The determinant scales the area of any measurable planar figure, not just squares. The absolute value of the determinant acts as a uniform area scaling factor for all regions under the linear transformation.

Conditions: The transformation is linear.; The figure is measurable and has finite area.; Use ordinary Euclidean area in standard orthonormal coordinates.

We multiply by 5 because the absolute value of the determinant of the transformation matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$ is 5. The determinant represents the area scaling factor of the linear transformation.

Conditions: The transformation matrix is $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.

We multiply by 5 because the absolute value of the determinant of the transformation matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$ is 5. The determinant represents the area scaling factor of the linear transformation.

Conditions: The transformation matrix is $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$.; The original area is known.; Use ordinary Euclidean area in standard orthonormal coordinates.