MHB 311.3.2.16 Find the determinant with variables a b c d e f g h i

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
$\tiny{311.3.2.16}$
Find the determinants where:
$\left|\begin{array}{rrr}a&b&c\\ d&e&f\\5g&5h&5i\end{array}\right|
=a\left|\begin{array}{rrr}e&f \\5h&5i\end{array}\right|
-b\left|\begin{array}{rrr}d&f \\5g&5i\end{array}\right|
+c\left|\begin{array}{rrr}d&e\\5g&5h\end{array}\right|=$

ok before I proceed on
just want see if this is correct
not sure why they thru the 5's in there
 
Physics news on Phys.org
This is correct, though I don't like the use of the "i." (Complex numbers and all.)

If you want to get rid of the 5's:
[math]\left | \begin{matrix} a & b & c \\ d & e & f \\ 5g & 5h & 5i \end{matrix} \right | = 5 \left | \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right | [/math]

-Dan
 
yeah, however I didn't know complex numbers were used in an matrix

$
a\left|\begin{array}{rrr}e&f \\5h&5i\end{array}\right|
-b\left|\begin{array}{rrr}d&f \\5g&5i\end{array}\right|
+c\left|\begin{array}{rrr}d&e\\5g&5h\end{array}\right|$
$=a(e5i-5hf)-b(d5i-5gf)+c(d5h-5ge)$
distirbute
$ae5i-a5hf-bd5i+b5gf+cd5d-c5ge$
rewrite
$5(aei-a5f-bdi+bgf+cdd-cge)$
hopefully,,, I quess the purpose of this was to show that 5 is a scaler
no book answer so not sure how to cross check this
 
thot I would throw in this question true or false

In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row

ok but isn't there just one form of RREF possoble if can be derived? there are multiple ways to reduce it but only one outcome
 
First, any numbers, including complex numbers, can appear in a matrix.

Second, row reduction of a matrix does NOT preserve its determinant. For example, factoring a number out of an entire row (or column) divides the determinant by that number. That is why, when Topsquark factored the "5"out of the bottom row, he multiplied the determinant by 5.

Swapping two rows, multiplies the determinant by -1.

Finally, adding a multiple of one row to another does not change the determinant.
 
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
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...
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...

Similar threads

Replies
3
Views
1K
Replies
2
Views
964
Replies
7
Views
1K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
3
Views
2K
Replies
14
Views
2K
Back
Top