find all minors and cofactors of [[ -2 -6 ][ 6 -3 ]]

[HopefulMii]

Hi! I have a few questions for my homework...it's due in a few days and i'm not sure how to do it. If you could also include explanations and your every steps that'd be really helpful too..thanks in advance!

Q1etermine all minors and cofactors of:

[ -2 -6 ]
[ 6 -3 ]

the answers i got were
M11: -3
C11: -3
M12: 6
C12: -6
M21: -6
C21:-6
M22:-2
C22:-2

However, I only have one try left (it's like..online math homework..and we have a limited amount of tries) and I really don't want to get it wrong...so if someone could check if I did it right ^^ That would be great!

Q2: Find the determinant of:

-3 0 0 -2 0
-2 0 3 0 0
0 -2 0 0 1
0 0 0 -1 3
0 -2 -2 0 0

I got a determinant of -66..but it says it's wrong and once again, I'm on my last try...


Q3:
If det of
a 1 d
b 1 e
c 1 f
is = 4

and det of
a 1 d
b 2 e
c 3 f
is = -3

than what is
a)det of
a 6 d
b 6 e
c 6 f

I got 24 for this one, and it's right, I got this by multiplying the determinent of the first matrix by 6

the problem, however, is
b) the det of
a 2 d
b 5 e
c 8 f
I'm not sure how to do this..

Q4:
-2 -2 -3
-2 3 -1
1 -2 3
How do you find the inverse for this?

and finally
Q5:

............

.............

How do you find it's det, it's cofactors, its adj and it's inverse?

I'm sorry this is alot of questions, I hope someone can help me ^^ Thanks in advance

Mii

 

[stapel]

If you could also include explanations and your every steps that'd be really helpful too..

It isn't reasonably feasible to provide lessons ("explanations" of the topic) within this environment, but we can provide links to online lessons. "Every step" is what you need to show. (We already know how to do this. What is needed now is to see where you are having problems. If looking at an example were sufficient, then the examples you saw in class and in your book would already have solved any difficulties.) So please reply with your work and reasoning, so that we know how to begin trying to help. Thank you!

Q1etermine all minors and cofactors of:

\(\displaystyle \left[\begin{array}{rr}-2&-6\\6&-3\end{array}\right]\)

To learn how to find minors and cofactors, try this list of lessons. (I only glanced at your answers, but they appear okay.)

Q2: Find the determinant of:

\(\displaystyle \left[\begin{array}{rrrrr}-3&0&0&-2&0\\-2&0&3&0&0\\0&-2&0&0&1\\0&0&0&-1&3\\0&-2&-2&0&0\end{array}\right]\)

To learn how to find determinants, try these lessons. (Your value is incorrect.)

Q3: If:

\(\displaystyle \left|\begin{array}{rrr}a&1&d\\b&1&e\\c&1&f\end{array}\right|\, =\, 4\)
\(\displaystyle \left|\begin{array}{rrr}a&1&d\\b&2&e\\c&3&f\end{array}\right|\, =\, -3\)

then find the value of:

\(\displaystyle \mbox{a) }\, \left|\begin{array}{rrr}a&6&d\\b&6&e\\c&6&f\end{array}\right|\)

I got 24 for this one, and it's right, I got this by multiplying the determinent of the first matrix by 6

the problem, however, is

\(\displaystyle \mbox{b) }\, \left|\begin{array}{rrr}a&2&d\\b&5&e\\c&8&f\end{array}\right|\)

I'm not sure how to do this..

The multipication by six worked because you multiplied a column by a value, which changed the matrix into what you needed, and changed the known determinantal value by what you'd multiplied. Use some of the other row- and column-operation tricks you've learned to figure out a way to relate the two givens to what you need to find.

Q4: Find the inverse of:

\(\displaystyle \left[\begin{array}{rrr}-2&-2&-3\\-2&3&-1\\1&-2&3\end{array}\right]\)

How do you find the inverse for this?

To learn how to find the inverse of a matrix, try some of these lessons

Q5:
\(\displaystyle \left[\begin{array}{rr}3e^{2t}&-6e^{2t}\\-6e^{3t}&3e^{2t}\end{array}\right]\)

How do you find it's det, it's cofactors, its adj and it's inverse?

Use the lessons in the links above to learn how to find determinants (that's what the "det" abbreviation means), cofactors, and inverses. To learn how to find adjoints (that's what the "adj" means), try these online lessons.

Once you have studied the lessons and have learned the basic terms and techniques, please attempt the exercises. If you get stuck, or if you are unsure of your answers, you will then be able to reply with a clear listing of your work and reasoning, and we can dig in and get starting working with you! :wink:

Eliz.

 

[HopefulMii]

Re: Linear Algebra

Hey,

I've tried doing what you listed above,

I still have no idea how to approach question 3.

For question 4, for some reason the first column is wrong, and I rechecked my calculations but still don't know what I did wrong...
here's what i did:
[attachment=2:12ps3wk0]Picture 005.jpg[/attachment:12ps3wk0]


For question 5, HEre's what I did to find the inverse (however, my answer is wrong...)
[attachment=1:12ps3wk0]Picture 002.jpg[/attachment:12ps3wk0]



Also, one last question I have (new one)
Q6: Find the determinant of

7 3 4 6 2
-5 0 2 9 9
6 0 0 9 9
x -6 -3 3 1
4 0 0 0 -1

her'es what i did:
[attachment=0:12ps3wk0]Picture 004.jpg[/attachment:12ps3wk0]

I hope my writing's not too messy ^^;
Thanks in advance!

Mii

 

[stapel]

I still have no idea how to approach question 3.

Hm... So you haven't studied systems of equations or linear combinations yet.... That's a problem....

I'm sorry, but I can't think of any way to "hint" (other than "do like you did for the first part, but with BOTH matrices", already provided) that doesn't simply give away the answer. Let's hope somebody more clever than I comes along!

For question 4, for some reason the first column is wrong, and I rechecked my calculations but still don't know what I did wrong...

In future, it will be helpful if you provide your reasoning. On a muddy graphic, it can be difficult to figure out what steps you may have taken when the results are unclear. I think your step was as follows (although I don't understand on what basis you "know" that some steps were "good" but others were "bad"...???):

[ -2 -2 -3 |  1  0  0 ]
[ -2  3 -1 |  0  1  0 ]  
[  1 -2  3 |  0  0  1 ]  

-R1 + R2; that is, [ 2  2  3 | 2  0  0 ] + [ -2  3 -1 | 0  1  0 ]

[ -2 -2 -3 |  1  0  0 ]
[  0  5  2 |  2  1  0 ]  
[  1 -2  3 |  0  0  1 ]

However, I have no idea what you did to the first row (R1) to get your result for R3...?

For question 5, HEre's what I did to find the inverse (however, my answer is wrong...)

What I could make out on this image seems... very complicated...? This is just a two-by-two matrix. Just do the diagonal multiplication, and subtract! :shock:

Also, one last question I have (new one)
Q6: Find the determinant of

[  7  3  4  6  2 ]
[ -5  0  2  9  9 ]
[  6  0  0  9  9 ]
[  x -6 -3  3  1 ] 
[  4  0  0  0 -1 ]

As you've learned, if you add a multiple of one row (or column) to another row (or column), the value of the determinant will not change. So it might be helpful, since the "x" won't simplify much, to make expansion along the first column or the fourth row a little less "painful" first. :wink:

|  7  3  4  6  2 |
| -5  0  2  9  9 |
|  6  0  0  9  9 |
|  x -6 -3  3  1 |
|  4  0  0  0 -1 |

. . . . .First, let's get that "x" up where it will be easy to keep track
. . . . .of, and do some rearranging to get the zeroes in a nice neat
. . . . .chunk. Let's switch R1 and R4, then R3 and R4, and then
. . . . .R2 and R3. This gives us:

|  x -6 -3  3  1 |
|  7  3  4  6  2 |
| -5  0  2  9  9 |
|  6  0  0  9  9 |
|  4  0  0  0 -1 |

. . . . .Since that's an odd number of swaps, the sign on the final
. . . . .value will need to be changed to get the right answer for
. . . . .the original determinant.

. . . . .The more zeroes we can create, the better. So let's do
. . . . .-2R3 + R2. Let's also multiply R4 by 1/3.

|  x -6 -3   3   1 |
| 17  3  0 -12 -16 |
| -5  0  2   9   9 |
|  2  0  0   3   3 |
|  4  0  0   0  -1 |

. . . . .Then do -3R4 + R3, and 4R4 + R2:

|   x -6 -3   3   1 |
|  25  3  0   0  -4 |
| -11  0  2   0   0 |
|   2  0  0   3   3 |
|   4  0  0   0  -1 |

. . . . .Then do -4R5 + R2 and 3R5 + R4:

|   x -6 -3   3   1 |
|   9  3  0   0   0 |
| -11  0  2   0   0 |
|  14  0  0   3   0 |
|   4  0  0   0  -1 |

Does that make the expansion a bit simpler?

Eliz.