Properties of Determinants
Properties of Determinants:

Multiplicative Property:
 For square matrices $A$ and $B$, $\text{det}(AB)=\text{det}(A)\times \text{det}(B)$.
 It signifies that the determinant of the product of two matrices is equal to the product of their individual determinants.

Transpose Property:
 For any square matrix $A$, $\text{det}(A)=\text{det}({A}^{T})$.
 The determinant of a matrix and its transpose are equal.

Inverse Property:
 If $A$ is an invertible matrix ($\text{det}(A)\mathrm{\ne}0$), then $\text{det}({A}^{1})=\frac{1}{\text{det}(A)}$.
 The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.

Scalar Multiplication:
 For a square matrix $A$ of order $n$ and scalar $k$, $\text{det}(kA)={k}^{n}\times \text{det}(A)$.
 The determinant of a matrix multiplied by a scalar is equal to the determinant of the original matrix raised to the power of the matrix's order, multiplied by the scalar.

Row/Column Operations:
 Performing elementary row or column operations on a matrix changes its determinant by a scalar factor but preserves its essential value.
 For example, interchanging rows or columns multiplies the determinant by $1$.
Significance and Applications:

Inversibility: A square matrix $A$ is invertible if and only if $\text{det}(A)\mathrm{\ne}0$.

Solvability of Systems: For a system of linear equations represented by a matrix $AX=B$, the system has a unique solution if and only if $\text{det}(A)\mathrm{\ne}0$.

Geometric Interpretation: The absolute value of the determinant represents the scaling factor in transformations and encodes information about the orientation change and volume scaling in space.