# Section 3.2 $\quad$ Properties of Determinants

## Properties of determinants

Let $A$ and $B$ be $n\times n$ matrices. Their determinants have following properties

1. $\det(A^T) = $ <br /><br /><br /><br />
- If $B = A_{r_i\leftrightarrow r_j}$ or $B = A_{c_i\leftrightarrow c_j}$ then <br /><br /><br /><br />
- If two rows (columns) of $A$ are equal, then <br /><br /><br /><br />
- If a row (column) of $A$ consists entirely of zeros, then <br /><br /><br /><br />
- If $B = A_{kr_i \to r_i}$ or $B = A_{kc_i\to c_i}$, then <br /><br /><br /><br />
- If $B = A_{kr_i+r_j \to r_j}$ or $B = A_{kc_i+c_j \to c_j}$, then <br /><br /><br /><br />
- If $A = [a_{ij}]$ is upper (lower) triangular, then <br /><br /><br /><br />

### Example 1

Find $\det(A)$ if
\begin{equation*}
  A =
  \left[
    \begin{array}{ccc}
      4 & 3 & 2 \\
      4 & -2 & 5 \\
      2 & 4 & 6 \\
    \end{array}
  \right]
\end{equation*}


```python
from sympy import *

A = Matrix([[4, 3, 2], [4, -2, 5], [2, 4, 6]]);

A.det()
```




    -130



## Properties of determinants (contd)

8.$\,$ If $E$ is an elementary matrix, then <br /><br /><br /><br />
9.$\,$ If $A$ is an $n\times n$ matrix, then $A$ is nonsingular if and only if <br /><br /><br /><br />
10.$\,$ If $A$ and $B$ are $n\times n$ matrices, then <br /><br /><br /><br />

### Example 2

Verify the property 10 using the matrices
\begin{equation*}
  A =\left[
    \begin{array}{cc}
      1 & 2 \\
      3 & 4 \\
    \end{array}
  \right]~~~
  B=
  \left[
    \begin{array}{cc}
      2 & -1 \\
      1 & 2 \\
    \end{array}
  \right]
\end{equation*}
<br /><br /><br /><br /><br /><br /><br />

### Example 3

- If $A$ is nonsingular. $\det(A^{-1}) = $ <br /><br /><br /><br />
- If $A$ and $B$ are $n\times n$ matrices, is $\det(AB) = \det(BA)$? <br /><br /><br /><br />

09f3ed00c64cdfe0a5df2e7e543bbd365fee5710
ipynb
Jupyter Notebook
Jupyter_Notes/Lecture09_Sec3-2_PropDet.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Jupyter_Notes/Lecture09_Sec3-2_PropDet.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Jupyter_Notes/Lecture09_Sec3-2_PropDet.ipynb
xiuquan0418/MAT341
2fb7ec4e5f0771f10719cb5e4a00a7ab07c49b59
MIT
Qwen/Qwen-72B
1. YES
2. YES