Algebra - Determinants: Properties, Minors, Co-factors

A Determinant is a scalar value associated with a square matrix, represented visually as a square array of numbers enclosed between two vertical bars, such as $|A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$. Unlike matrices, a determinant must always have an equal number of rows and columns ($n \times n$).

The Minor ($M_{ij}$) of an element $a_{ij}$ is the determinant obtained by deleting the $i^{th}$ row and $j^{th}$ column in which the element $a_{ij}$ lies. Visually, if you cross out the horizontal and vertical lines intersecting at $a_{ij}$, the remaining elements form the sub-matrix for the minor.

The Co-factor ($A_{ij}$ or $C_{ij}$) is the signed version of the minor, calculated as $A_{ij} = (-1)^{i+j} M_{ij}$. The signs follow a checkerboard pattern across the determinant grid, starting with a $+$ sign at the top-left corner $(a_{11})$ and alternating signs for adjacent elements.

Expansion of a Determinant can be performed along any row or any column. The value of the determinant is the sum of the products of elements of any row (or column) with their corresponding co-factors. For a $3 \times 3$ determinant, this is often visualized as taking the top row elements and multiplying them by the $2 \times 2$ determinants that remain after blocking their respective rows and columns.

Property of Identical Rows/Columns: If any two rows or columns of a determinant are identical or proportional, the value of the determinant is zero. Visually, if you see two horizontal lines of numbers that are exactly the same, you can immediately conclude $|A| = 0$.

Scalar Multiplication Property: If every element of a row (or a column) of a determinant is multiplied by a constant $k$, then the value of the determinant gets multiplied by $k$. This differs from matrices where a scalar multiplies every single element; in determinants, it only affects one row or column at a time.

Property of Invariance: The value of a determinant remains unchanged if we add a multiple of one row to another row (or one column to another column), denoted as $R_i \rightarrow R_i + kR_j$ or $C_i \rightarrow C_i + kC_j$. This allows for simplifying complex determinants into upper triangular forms where most elements below the diagonal are zero.

Area of a Triangle: Determinants provide a visual geometric application 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 zero, it indicates that the three points are collinear (lying on a single straight line).