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The determinant represents the signed area scaling factor of the linear transformation defined by the matrix. Geometrically, the columns of the matrix form two vectors that span a parallelogram.

Conditions: A is $a 2\times 2$ matrix.; The columns of A are interpreted as two vectors in the plane.; Use ordinary Euclidean area in standard orthonormal coordinates.

The determinant represents the signed area scaling factor of the linear transformation defined by the matrix. Geometrically, the columns of the matrix form two vectors that span a parallelogram.

Conditions: A is $a 2\times 2$ matrix.; The columns of A are interpreted as two vectors in the plane.; 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.