# 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 $\displaystyle \left[\begin{matrix} 4 & -3 & 5 \\ 5 & 2 & 0 \\ 2 & 0 & 4 \end{matrix} \right]$

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

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 $\displaystyle \left[\begin{matrix} 4 & -3 & 5 \\ 5 & 2 & 0 \\ 2 & 0 & 4 \end{matrix} \right]$

by adding -2.5 times the first row to the second gives: $\displaystyle \left[\begin{matrix} 4 & -3 & 5 \\ 0 & -5.75 & -6.25 \\ 0 & 1.5 & 1.5 \end{matrix} \right]$
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
$\displaystyle \left[\begin{matrix}4 & -3 & 5 \\0 & -5.75 & -6.25 \\2 & 0 & 4 \end{matrix} \right]$
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.