MIT18.06 Linear Algebra (18)

lec18

$$\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=a d-b c$$

determinate

• $\det I=1$
• Exchange rows, reverse sign of $\det$ , so $\det P=1$ or $-1$
• $\begin{vmatrix} ta & tb\\ c & d\end{vmatrix} =t\begin{vmatrix} a & b\\ c & d\end{vmatrix}$
• $\begin{vmatrix} a+a' & b+b'\\ c & d\end{vmatrix} =\begin{vmatrix} a & b\\ c & d\end{vmatrix} +  \begin{vmatrix} a' & b'\\ c & d\end{vmatrix}$
• If two rows are equal, $\det=0$
• substract $l*row$ from rows, $\det$ remain same.

triangular

$$\begin{vmatrix} d_1 & * & *\\ 0 & d_2 & * \\ 0 & 0 & d_3 \end{vmatrix} = d_1d_2d_3$$

if the matrix is singular, $\det = 0$ .

• $\det AB=(\det A)(\det B)$ , $\det A^{-1} = \frac{1}{\det A}$ ( $A$ is invertible)
• $\det 2A=2^n\det A$
• $\det A^\top =\det A$