Determinants - Minors, co-factors and applications of determinants in finding the area of a triangle

Minor of an element $a_{ij}$: The minor $M_{ij}$ is the determinant of the sub-matrix obtained by deleting the $i^{th}$ row and $j^{th}$ column in which the element $a_{ij}$ lies. Visually, you can imagine crossing out the vertical and horizontal lines passing through $a_{ij}$ and looking at the remaining block of numbers.

Cofactor of an element $a_{ij}$: The cofactor $A_{ij}$ is defined as $(-1)^{i+j}$ times the minor $M_{ij}$. Visually, this follows a checkerboard sign pattern starting with $+$ at the top-left corner: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$.

Expansion of Determinant: The value of a determinant is the sum of products of elements of any row (or column) with their corresponding cofactors. For example, expanding along Row 1: $\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$. Conversely, if elements of one row are multiplied with cofactors of a different row, the sum is always zero.

Area of a Triangle: The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by half the absolute value of the determinant formed by the coordinates. Because area is a scalar magnitude, we always take the positive value of the result.

Condition for Collinearity: If three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are collinear, they lie on a single straight line and do not form a triangle. Geometrically, this means the area of the triangle formed by them must be zero, leading to the determinant of their coordinates being zero.

Equation of a Line: To find the equation of a line passing through two given points $(x_1, y_1)$ and $(x_2, y_2)$, we take an arbitrary point $(x, y)$ on the line and apply the condition of collinearity for these three points. This results in a linear equation in $x$ and $y$.