> [!NOTE] Definition (Determinant) > For any [[Real Square Matrices|square matrix]] $A\in \mathbb{R}^{n\times n},$ there is an associated scalar $\det A$ called the **determinant** of $A$ which, regarded as a function, $\begin{align} \mathbb{R}^{n\times n} &\to \mathbb{R} \\ A &\mapsto \det A \end{align}$has the following properties with respect to [[Elementary Row Operation is Equivalent to Pre-Multiplying by Elementary Matrix|elementary matrices]]: > >1. $\det(S_{ij}A)=-\det A$ >2. $\det(M_{j}(\lambda)A)=\lambda \det A$ >3. $\det(A_{ij}(\lambda)A)=\det A$ >4. $\det I_{n}=1$ # Properties > [!NOTE] Theorem (Properties of the determinant) > The determinant $\det A$ of a matrix $A$ satisfies the following properties: >1. If a row of $A$ is zero, then $\det A=0.$ >2. If two rows of $A$ are identical, then $\det A=0.$ >3. If $A$ is a diagonal matrix, that is $A_{ij}=0$ whenever $i\neq j$, then $\det A=a_{11}a_{22}\cdots a_{nn}.$ >4. If $A$ is an upper triangular matrix, that is $A_{ij}=0$ whenever $i>j$ then $\det A=$ $a_{11}a_{22}\cdots a_{nn}.$ >5. If $A$ is a lower triangular matrix, that is $A_{ij}=0$ whenever $i<j$, then $\det A=$ $a_{11}a_{22}\cdots a_{nn}.$ >Proof.