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Geometrically, the determinant of $a 2\times 2$ matrix represents the factor by which the linear transformation scales areas. Specifically, it is the signed area of the parallelogram formed by the matrix's column vectors.

Conditions: The matrix is $2\times 2$.; The transformation is linear.; Use ordinary Euclidean area in standard orthonormal coordinates.

Geometrically, the determinant of $a 2\times 2$ matrix represents the factor by which the linear transformation scales areas. Specifically, it is the signed area of the parallelogram formed by the matrix's column vectors.

Conditions: The matrix is $2\times 2$.; The transformation is linear.; Use ordinary Euclidean area in standard orthonormal coordinates.

The determinant is computed using the standard $2\times 2$ rule $ad - bc$. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, this means multiplying the main diagonal entries ($3 \cdot 2$) and subtracting the product of the off-diagonal entries ($1 \cdot 1$).

Conditions: The matrix is $2\times 2$.; Entries are real numbers.

The determinant is computed using the standard $2\times 2$ rule $ad - bc$. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, this means multiplying the main diagonal entries ($3 \cdot 2$) and subtracting the product of the off-diagonal entries ($1 \cdot 1$).

Conditions: The matrix is $2\times 2$.; Entries are real numbers.

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.