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The matrix is $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{bmatrix}$. This is derived by tracking the standard basis vectors: $\hat{i}$ moves to $(0, 0, -1)$, $\hat{j}$ remains stationary at $(0, 1, 0)$, and $\hat{k}$ swings to $(1, 0, 0)$.

Conditions: Rotation is 90 degrees around the y-axis.; Working in a right-handed 3D coordinate system.

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.

When computing the composition $A \circ B$, the matrix B represents the first transformation. Its columns are the images of the standard basis vectors under B.

Conditions: A and B are 3x3 matrices representing linear transformations.; The goal is to find the matrix representation of the composite transformation.