I cant explain why these matrix row transformations are wrong

35 viewed last edited 7 months ago

I was helping someone find the determinant of the matrix:

[math] \left[ \begin{array}{ccc} 1 & -a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{array} \right] [/math]

Apparently, most students in the class applied the following row transformations:

R1 = R1 - R2 and

R2 = R2 - R3 and

R3 = R3 - R1

to get

[math] \left[ \begin{array}{ccc} 0 & (a-b) & (a^2-b^2) \\ 0 & (b-c) & (b^2-c^2) \\ 0 & (c-a) & (c^2-a^2) \\ \end{array} \right] [/math]

The determinant of this matrix is 0. However, the determinant of the original matrix is (a-b)(b-c)(c-a).

What's wrong with the row transformations applied?

Vivekanand Vellanki

Row transformations have to be applied one at a time. In this case, if the three transformations were applied one after the other, they would look like this:

R1 = R1 - R2; R2 = R2; R3 = R3 after the first transformation; and

R1 = R1 - R2; R2 = R2 - R3; R3 = R3 after the second transformation; and

R1 = R1 - R2; R2 = R2 - R3; R3 = R3 - (R1 - R2) after the third transformation

I see that teachers apply multiple such transformations at the same time. While this is possible and saves time in class, it has to be done with caution.

I am working on figuring out simple rules that inform if multiple row transformations can be applied at the same time.