find the determinant where M has no value

[atw]

I need some help please. I need to find the determinant where m is just a letter and has no know numerical value.

3 -6 4
9 2 3
6 -7 M

I don't know where to begin. I multiplied the first row by -3 and added it to the second row. I'm not sure if I'm doing this correctly. Please help me!!!!

Thanks,

atw

 

[fasteddie65]

|3 -6 4|
|9 2 3|
|6 -7 M|

That's a good start.

You can also use the diagonal method:

Copy the first two columns next to the determinant.
3 -6 4 3 -6
9 2 3 9 2
6 -7 M 6 -7

3 • 2 • M + (-6) • 3 • 6 + 4 • 9 * (-7) - 6 • 2 • 4 - (-7) • 3 • 3 - M • 9 • (-6) -

Notice that the upper down diagonal products are added, while the lower up diagonal products are subtracted.

 

[soroban]

Hello, atw!

You're right . . . I copied it incorrectly from my notes.

Could you provide the original wording of the problem?
As written, it doesn't make any sense.

Find the determinant where \(\displaystyle m\) is just a letter ] . It's not a variable?
and has no known numerical value. . What has no known numerical value?

. . \(\displaystyle \left|\begin{array}{ccc}3 & \text{-}6 & 4 \\9 & 2 & 3 \\ 6 & \text{-}7 & m \end{array}\right|\)

\(\displaystyle m\) will always have a known numerical value . . . \(\displaystyle \text{even }0,\:\pi,\:i,\text{ or }\infty.\)
. . And, of course, so will the determinant.
So what are they asking for?

If they want the value of \(\displaystyle m\) which makes the determinant equal to zero,
. . why didn't they/you say so?
This has nothing to do with "no known numerical value".


Fasteddie showed you one way to evaluate the determinant.
Are you familiar with the "cofactor method"?

\(\displaystyle \left|\begin{array}{ccc}3 & \text{-}6 & 4 \\ 9 & 2 & 3 \\ 6 & \text{-}7 & m \end{array}\right| \;=\;3\left|\begin{array}{cc}2 & 3 \\ \text{-}7 & m\end{array}\right| - (\text{-}6)\left|\begin{array}{cc}9&3\\6&m\end{array}\right| + 4\left|\begin{array}{cc}9 & 2 \\ 6 & \text{-}7\end{array}\right|\)

. . . . . . . . . .\(\displaystyle = \;3(2m+21) + 6(9m - 18) + 4(-63-12)\)

. . . . . . . . . .\(\displaystyle = \;6m + 63 + 54m - 108 - 252 - 48\)

. . . . . . . . . .\(\displaystyle = \;60m - 345\)


\(\displaystyle \text{If }60m-345 \:=\:0\text{, then: }\:m \:=\:\frac{345}{60} \:=\:\frac{23}{4} \:=\:5\tfrac{3}{4}\)

 

[atw]

aren't you suppose to add the 6m and 54m and it becomes 60m? I'm confused with the answer? Please help me, thanks.

 

[mmm4444bot]

.. I'm confused ... help me ...

If you were to find initiative to verify 345/60 (m = 23/4), then you would know whether or not 54m was dropped, as it appears to have been.

Do you know how to calculate the determinant after substituting 23/4 for m?

 

[Denis]

Well, if you're confused (understandable, since you're learning this),
why don't you start by setting 'em up as equations:
3a - 6b = 4 [1]
9a + 2b = 3 [2]
6a - 7b = m [3]

Now get a and b from [1] and [2], then get m from [3].
Then you'll have something to follow while solving from matrix....kapish?

 

[atw]

Yes I know how to calculate the determinant of 23/4 and when I put it in the calculator I got 0

 

[mmm4444bot]

Yes I know how to calculate the determinant of 23/4 and when I put it in the calculator I got 0

Does this tell you that 23/4 is the correct value for m? Why are you confused?