Determinant of row interchange proof

fackert
Messages
3
Reaction score
0
Need a lot of help here guys.

I need to prove that for an nxn matrix A, if i interchange two rows to obtain B, then det=-detA

I have proved my basis (below), but I'm stuck on the hard part, the induction (which I'm required to do). I understand the steps of induction, but i don't know how to do it for this case.

What i have so far:

Let A be an nxn matrix.
Basis n=2
Then detA=a(11)a(22) - a(12)a(21)
Now let B be the matrix obtained by interchanging rows 1 and 2
Then detB=a(21)a(12) - a(22)a(11)
=-detA
So true for an arbitary 2x2 matrix.

(induction)
Assume true for n=k
For a kxk matrix...?
 
Physics news on Phys.org
Hey fackert.

Try introducing a pre-multiplication matrix that swaps the rows and then use the relationship that det(A*B) = det(A)*det(B) where B is your original matrix and A is a matrix transformation that swaps the rows.
 
my notes on determinants are on page 62ff of these notes:

http://www.math.uga.edu/%7Eroy/4050sum08.pdf
 
Last edited by a moderator:

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Applying change of basis to vector b_1 doesn't give b'_1?
Replies
8
Views
2K
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
B Is it invalid to redefine the sgn function in this way in a proof?
Replies
15
Views
4K
B Proof of elementary row matrix operation.
Replies
9
Views
3K
I Proof by induction of block diagonal decomposition of a matrix
Replies
4
Views
1K
I Proof that if T is Hermitian, eigenvectors form an orthonormal basis
Replies
3
Views
3K
I How can I convince myself that I can find the inverse of this matrix?
Replies
34
Views
3K
I Explanation of a Line of a proof in Axler Linear Algebra Done Right 3r
Replies
4
Views
2K
I Question regarding fundamental region of a lattice
Replies
20
Views
2K
MHB Solving the Matrix Transformation: $B \to C$
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top