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

[karush]

$\tiny{311.3.2.16}$
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

 

[topsquark]

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:
\(\displaystyle \left | \begin{matrix} a & b & c \\ d & e & f \\ 5g & 5h & 5i \end{matrix} \right | = 5 \left | \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right | \)

-Dan

 

[karush]

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

 

[karush]

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

 

[HOI]

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.