Finding the Determinant of a Transposed Matrix with Column Swapping

[nlsherrill]

Homework Statement

From "Introduction to Linear Algebra with applications" by Defranza. Ch.1 section 1.6 prob 30.

Let the matrix

a b c
A = d e f
g h i

Where det(A)= 10

Find

a g d
det b h e
c i f

(sorry I didnt see how to write a matrix in latex, but it should be pretty clear what I mean)

**The matrix looks OK when I am typing it out, but when I submitted the thread it messed with it and it looks really messed up**

Its basically a 3x3 matrix with elements(from a11 to a33) a, b, c, d, e, f, g, h, i

Homework Equations

The Attempt at a Solution

So I noticed that the matrix they are asking to find is the transpose of A, plus the 3rd column has been swapped with the 2nd. I haven't read anything about column swapping, so I am not sure if that would alter the the determinate of a matrix like a row swap would. I know det(Atranspose)=det(A), so I figure the answer to this is either 10 or -10(due to the column swap)

Any advice/hints would be appreciated. I also searched in my book and on google and did not find any conclusive results on column swapping as far as elementary row operations go.

 

[fzero]

What operation on the original matrix corresponds to the column swap on the transpose?

 

[nlsherrill]

What operation on the original matrix corresponds to the column swap on the transpose?

Ah HA!

so the answer must be -10 correct?

 

[fzero]

Yes, each swap of adjacent rows or columns introduces a minus sign in the determinant.