Properties of the Determinant

Determinant for Rows or Columns with Zero

Let $A$ be a square matrix with a least one row or column consisting of only zeroes. Then $det\ A=0$.

Example

$A=\begin{bmatrix}1&2\\0&0\end{bmatrix}\\ det\ A=1*0-2*0=0$

Determinant for Similar Rows or Columns

Let $A$ be a square matrix with one row or column exactly the same as a second row or column respectively. Then $det\ A=0$.

Example

$A=\begin{bmatrix}3&4\\3&4\end{bmatrix}\\ det\ A=3*4-3*4=0$

Determinant for Multiplied Matrices

Let $A$ and $B$ be square matrices of similar size. Then $det\ AB=det\ A*det\ B$.

Example

$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\\ B=\begin{bmatrix}8&6\\5&7\end{bmatrix}\\ det\ A=1*4=2*3=-2\\ det\ B=8*7-6*5=26\\ det\ A*\det\ B=-2*26=-52\\ AB=\begin{bmatrix}18&20\\44&46\end{bmatrix}\\ det\ AB=18*46-20*44=-52\\ \therefore det\ AB=det\ A*det\ B$

Determinant for Singular Matrices

A matrix $A$ is said to be singular if and only if $det\ A=0$.

Determinant for Inverse Matrices

Let $A$ be a square matrix and let $A^{-1}$ be it's inverse. Then .

Example

$A=\begin{bmatrix}10&4\\7&3\end{bmatrix}\\ det\ A=10*3-4*7=2\\ A^{-1}=\begin{bmatrix}\frac32&-2\\-\frac72&5\end{bmatrix}\\ det\ A^{-1}=(\frac32\ *5)-(-2*-\frac72\ )=\frac12\\ \therefore det\ A^{-1}=\frac{1}{det\ A}$

Determinants from Row Operations

Row Switch

Let $A$ be a square matrix and let $B$ be the new matrix created when two rows or columns have been switched. Then $det\ B=-det\ A$.

Example

$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\\ R_1\longleftrightarrow R_2\\ \begin{bmatrix}3&4\\1&2\end{bmatrix}=B\\ det\ A=1*4-2*3=-2\\ det\ B=3*2-1*4=2\\ \therefore det\ B=-det\ A$

Row Multiplication

Let $A$ be a square matrix and let $B$ be the new matrix created when one row or column of matrix $A$ is multiplied by a scalar $c$. Then $det\ B=c*det\ A$.

Example

$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\\ 2R_2\longrightarrow R_2\\ \begin{bmatrix}1&2\\\color{red}6&\color{red}8\end{bmatrix}=B\\ det\ A=1*4-2*3=-2\\ det\ B=1*8-2*6=-4\\ \therefore det\ B=2*det\ A$

Row Addition

Let $A$ be a square matrix and let $B$ be the new matrix created when one row or column of matrix $A$ is multiplied by a scalar $c$ and added to another row. Then $det\ B=det\ A$.

Example

$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}\\ 5R_1+R_2\longrightarrow R_2\\ \begin{bmatrix}1&2\\\color{red}8&\color{red}{14}\end{bmatrix}=B\\ det\ A=1*4-2*3=-2\\ det\ B=1*14-2*8=-2\\ \therefore det\ B=det\ A$