Determinants - Determinant of a square matrix (up to 3 x 3 matrices)

Determinant Definition: Every square matrix $A = [a_{ij}]$ is associated with a scalar value called its determinant, denoted by $|A|$ or $det(A)$. For a $1 \times 1$ matrix $A = [a]$, the determinant is simply $|A| = a$. It is important to remember that only square matrices have determinants.

Determinant of a $2 \times 2$ Matrix: For a matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is calculated by the 'cross-multiplication' method. Visually, imagine drawing two diagonal arrows: one from $a$ to $d$ (principal diagonal) and one from $b$ to $c$ (off-diagonal). The value is the product of the first diagonal minus the product of the second: $ad - bc$.

Expansion of a $3 \times 3$ Matrix: The determinant of a $3 \times 3$ matrix is found by breaking it down into smaller $2 \times 2$ determinants (minors). You can expand along any row or column. For example, expanding along the first row involves taking each element $a_{1j}$, multiplying it by its $2 \times 2$ minor (the determinant remaining after deleting the row and column of $a_{1j}$), and applying a specific sign.

Sign Convention (Checkerboard Pattern): When expanding a $3 \times 3$ determinant, the sign of each term is determined by its position $(i, j)$ using the formula $(-1)^{i+j}$. Visually, this creates a checkerboard pattern of signs: $\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$. This pattern ensures that adjacent elements always have opposite signs.

Singular vs. Non-Singular Matrices: A square matrix $A$ is called 'Singular' if its determinant $|A| = 0$. Visually, this means the matrix maps a space into a lower dimension (e.g., a 2D plane into a 1D line). If $|A| \neq 0$, the matrix is 'Non-singular' and possesses an inverse.

Area of a Triangle: Determinants provide a geometric tool to find the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. If the calculated determinant value is $0$, it visually confirms that the three points are collinear, meaning they lie on the same straight line and do not form a triangle.

Minors and Cofactors: The minor $M_{ij}$ of an element $a_{ij}$ is the determinant of the sub-matrix obtained by deleting the $i^{th}$ row and $j^{th}$ column. The cofactor $A_{ij}$ is the minor with the positional sign applied, calculated as $A_{ij} = (-1)^{i+j} M_{ij}$.

Properties of Determinants: (1) The determinant of a matrix and its transpose are equal: $|A| = |A^T|$. (2) If any two rows (or columns) of a determinant are identical, the value of the determinant is zero. (3) If all elements of a row or column are zero, the determinant is zero.