Determinants & Their Properties

To every square matrix $A$ we attach a single number, its determinant, written $\det A$ or $|A|$. It is defined only for square matrices and answers one decisive question: is $A$ invertible? ($|A|\ne 0$ means yes.) Geometrically it measures how the linear map $A$ scales area (in $2\times2$) or volume (in $3\times3$), and its sign records whether orientation is preserved.

Determinant of order 1, 2 and 3

For a $1\times1$ matrix $A=[a]$, $|A|=a$. For a $2\times2$ matrix,

The number is the product of the leading (main) diagonal minus the product of the off-diagonal. For a $3\times3$ matrix we expand along any one row or column, multiplying each entry by its signed minor (cofactor). Expanding along the first row:

The alternating $+,-,+$ signs follow the checkerboard pattern $(-1)^{i+j}$. A vital practical point: the value does not depend on which row or column you expand along, so always choose the line with the most zeros to save arithmetic.

Properties of determinants

These properties let you simplify a determinant before expanding — often reducing a messy $3\times3$ to almost nothing. Let $A$ be $n\times n$.

• Property 1 (Reflection): $|A^{T}|=|A|$. Rows and columns play identical roles, so any property true for rows holds for columns.
• Property 2 (Interchange): swapping two rows (or two columns) multiplies the determinant by $-1$.
• Property 3 (All-zero line): if every entry of some row or column is $0$, then $|A|=0$.
• Property 4 (Identical/proportional lines): if two rows (or columns) are equal, or one is a scalar multiple of another, then $|A|=0$.
• Property 5 (Scalar / factor): if every entry of one row (or column) carries a common factor $k$, that $k$ can be taken outside the determinant. Hence for $n\times n$, $|kA|=k^{n}|A|$.
• Property 6 (Sum-splitting): if each entry of one row is written as a sum, the determinant splits into a sum of two determinants, e.g. $\begin{vmatrix}a_{1}+x&b_{1}&c_{1}\\a_{2}+y&b_{2}&c_{2}\\a_{3}+z&b_{3}&c_{3}\end{vmatrix}=\begin{vmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{vmatrix}+\begin{vmatrix}x&b_{1}&c_{1}\\y&b_{2}&c_{2}\\z&b_{3}&c_{3}\end{vmatrix}.$
• Property 7 (Row-operation invariance): adding to one row a multiple of another row leaves the determinant unchanged, i.e. $R_{i}\to R_{i}+kR_{j}$ (or $C_{i}\to C_{i}+kC_{j}$) does not alter $|A|$. This is the single most useful tool for creating zeros and for proving identities.
• Multiplicative law: $|AB|=|A|\,|B|$, and a triangular matrix has determinant equal to the product of its diagonal entries.

A determinant in which every entry of a row or column is the same as another after an operation collapses to $0$, which is why row operations are the engine behind every "prove that the determinant equals …" question.

Deeper Insight — why the row operation $R_i\to R_i+kR_j$ changes nothing: By sum-splitting (Property 6), $|A \text{ with } R_i\to R_i+kR_j|$ equals the original determinant plus a second determinant whose row $i$ is $k$ times row $j$. That second determinant has two proportional rows, so by Property 4 it is $0$ — leaving the original value untouched. Understanding this one chain (split $\to$ proportional $\to$ zero) explains all the simplification you will do in this chapter: you are free to add multiples of rows to one another to manufacture zeros, and the answer never moves.