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Algebraically, for a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. Geometrically, $ad$ captures the primary rectangular bounds, while subtracting $bc$ corrects for overlapping triangular regions created by off-diagonal shearing components.

Conditions: Matrix is 2x2; Entries are real numbers

Algebraically, for a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. Geometrically, $ad$ captures the primary rectangular bounds, while subtracting $bc$ corrects for overlapping triangular regions created by off-diagonal shearing components.

Conditions: Matrix is 2x2; 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.

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.

$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

$A 2\times 2$ transformation matrix maps the standard basis vectors to its own columns. Specifically, the first column of the matrix is the image of the vector $[1, 0]^T$, and the second column is the image of the vector $[0, 1]^T$.

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

Each column of $a 2\times 2$ matrix can be read as the coordinates of a vector in the 2D plane. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, the first column $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ represents the vector from the origin to the point (3,1), and the second column $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ represents the vector from the origin to the point (1,2).

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian coordinates.

Each column of $a 2\times 2$ matrix can be read as the coordinates of a vector in the 2D plane. For the matrix $A = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, the first column $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ represents the vector from the origin to the point (3,1), and the second column $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ represents the vector from the origin to the point (1,2).

Conditions: The matrix is $2\times 2$.; Working in standard Cartesian 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.

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.

A 3D linear transformation is fully determined by tracking where the standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land. The coordinates of these three transformed vectors are recorded as column vectors to form a 3x3 matrix.

Conditions: Working in three-dimensional Cartesian space; Using the standard basis vectors aligned with x, y, and z axes; The transformation is linear (preserves grid lines parallel/evenly spaced and fixes origin)

A 3D linear transformation is fully determined by tracking where the standard basis vectors ($\hat{i}$, $\hat{j}$, and $\hat{k}$) land. The coordinates of these three transformed vectors are recorded as column vectors to form a 3x3 matrix.

Conditions: Working in three-dimensional Cartesian space; Using the standard basis vectors aligned with x, y, and z axes; The transformation is linear (preserves grid lines parallel/evenly spaced and fixes origin)

Starting from $Av = \lambda v$, we rewrite the right side as $(\lambda I)v$ and move all terms to one side to get $(A - \lambda I)v = 0$. Since we seek non-zero solutions for $v$, the matrix $(A - \lambda I)$ must squash space into a lower dimension (have a non-trivial null space).

Conditions: $v$ is a non-zero eigenvector; $A$ is a square matrix; $I$ is the identity matrix

Starting from $Av = \lambda v$, we rewrite the right side as $(\lambda I)v$ and move all terms to one side to get $(A - \lambda I)v = 0$. Since we seek non-zero solutions for $v$, the matrix $(A - \lambda I)$ must squash space into a lower dimension (have a non-trivial null space).

Conditions: $v$ is a non-zero eigenvector; $A$ is a square matrix; $I$ is the identity matrix