# Thread: confusion involving method to compute determinant

1. ## confusion involving method to compute determinant

1) Do all square matrices have a unique determinant?
2) If the answer to the above question is yes, then what part of the following procedure am I over looking?

I am trying to use "computation via reduction to a triangular form" to find the determinant of $\left[\begin{matrix}
4 & -3 & 5 \\
5 & 2 & 0 \\
2 & 0 & 4
\end{matrix} \right]
$

by adding -2.5 times the first row to the second gives: $\left[\begin{matrix}
4 & -3 & 5 \\
0 & -5.75 & -6.25 \\
0 & 1.5 & 1.5
\end{matrix} \right]
$

Here's my confusion. To eliminate the first 1.5 I can either add 1/2 the first row to the third or -5.75/1.5 the second row to the third. Depending on which route I take the last element in the third row will be differnt, and when I multiply the numbers along the diagonal I will get 60 or 72 for the determinant.

2. Originally Posted by superdude
1) Do all square matrices have a unique determinant?
2) If the answer to the above question is yes, then what part of the following procedure am I over looking?

I am trying to use "computation via reduction to a triangular form" to find the determinant of $\left[\begin{matrix}
4 & -3 & 5 \\
5 & 2 & 0 \\
2 & 0 & 4
\end{matrix} \right]
$

by adding -2.5 times the first row to the second gives: $\left[\begin{matrix}
4 & -3 & 5 \\
0 & -5.75 & -6.25 \\
0 & 1.5 & 1.5
\end{matrix} \right]
$
Well, you also subtracted 1/2 of the first line from the third line!

Here's my confusion. To eliminate the first 1.5 I can either add 1/2 the first row to the third or -5.75/1.5 the second row to the third. Depending on which route I take the last element in the third row will be differnt, and when I multiply the numbers along the diagonal I will get 60 or 72 for the determinant.
If you "add 1/2 the first row to the third" you get
$\left[\begin{matrix}4 & -3 & 5 \\0 & -5.75 & -6.25 \\2 & 0 & 4
\end{matrix} \right]$

Which is NOT triangular because you have added (1/2)(4)= 2 to the 0 that was in the first column of the third line.

That is why you should always work with the line that has only "0"s to the left of the pivot.