# Properties of Determinants

### Properties of Determinants:

1. Multiplicative Property:

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

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

• If $A$ is an invertible matrix ($\text{det}\left(A\right)\mathrm{\ne }0$), then $\text{det}\left({A}^{-1}\right)=\frac{1}{\text{det}\left(A\right)}$.
• The determinant of the inverse of a matrix is the reciprocal of the determinant of the original matrix.
4. Scalar Multiplication:

• For a square matrix $A$ of order $n$ and scalar $k$, $\text{det}\left(kA\right)={k}^{n}×\text{det}\left(A\right)$.
• 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.
5. 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:

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

2. 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 $\text{det}\left(A\right)\mathrm{\ne }0$.

3. 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.