I have a matrix 4x4 :
\begin{matrix} 2 & 1 & 2 & 8 \\ -1 & 0 & -2 & 4 \\ 4 & 2 & 8 & 2 \\ 2 & 2 & 2 & 2 \end{matrix}
then I want to find determinan using lower triangular,which is the pattern of lower triangular :
\begin{matrix} l11 & 0 & 0 & 0 \\ l21 & l22 & 0 & 0 \\ l31 & l32 & l33 & 0 \\ l41 & l42 & l43 & l44 \end{matrix}
here my steps that I was try :
Step 1 : change 8,4,2 to 0
\begin{matrix} 2 & 1 & 2 & \color{blue} 8 \\ -1 & 0 & -2 & \color{blue} 4 \\ 4 & 2 & 8 & \color{blue} 2 \\ 2 & 2 & 2 & 2 \end{matrix}
by :
-4 x Row 4+Row 1
-2 x Row 4+Row 2
-1 x Row 4+Row 3
Result 1 :
\begin{matrix} -6 & -7 & -6 & \color{purple} 0 \\ -5 & -4 & -6 & \color{purple} 0 \\ 2 & 0 & 6 & \color{purple} 0 \\ 2 & 2 & 2 & 2 \end{matrix}
Step 2 : change -6,-6 to 0
\begin{matrix} -6 & -7 & \color{blue}-\color{blue}6 & 0 \\ -5 & -4 & \color{blue}-\color{blue}6 & 0 \\ 2 & 0 & 6 & 0 \\ 2 & 2 & 2 & 2 \end{matrix}
By :
1x Row 3 + Row 1
1x Row 3 + Row 2
Result 2 :
\begin{matrix} -4 & -7 & \color{purple}0 & 0 \\ -3 & -4 & \color{purple}0 & 0 \\ 2 & 0 & 6 & 0 \\ 2 & 2 & 2 & 2 \end{matrix}
Step 3 : change -7 to 0
\begin{matrix} -6 & \color{blue}-\color{blue}7 & 0 & 0 \\ -3 & -4 & 0 & 0 \\ 2 & 0 & 6 & 0 \\ 2 & 2 & 2 & 2 \end{matrix}
By :
$\frac{-7}{4}$ x Row 2 + Row 1
Result 3 :
\begin{matrix} 1 \frac{1}{4} & \color{purple}0 & 0 & 0 \\ -3 & -4 & 0 & 0 \\ 2 & 0 & 6 & 0 \\ 2 & 2 & 2 & 2 \end{matrix}
and then I multiplies the diagonal :
\begin{matrix} \color{red}1\frac{1}{4} & 0 & 0 & 0 \\ -3 & \color{red}-\color{red}4 & 0 & 0 \\ 2 & 0 & \color{red}6 & 0 \\ 2 & 2 & 2 & \color{red}2 \end{matrix}
1 $\frac{1}{4}$ x -3 x 6 x 2 = 240
Why I get 240 ? it's should -60
Anyone can correct my matrix or steps ?
Just the last step is wrong. What is?
$$1.25\cdot\left(-4\right)\cdot6\cdot2$$
And $-6$ in the beginning of step 3 should've been $-4$ but that doesn't matter, you didn't carry this error to the result of the step.