# Homework Help: Property of Determinants Answers Check

1. May 9, 2013

### muzziMsyed21

1. The problem statement, all variables and given/known data
Let A and P be square matrices of the same size with P invertible, Prove detA=det(P-1AP)

2. Relevant equations
Suppose that A and B are square matrices of the same size. Then det(AB)=det(A)det(B)

3. The attempt at a solution

detA=det(P-1AP)
detA=det(P-1PA)
detA=det(IA)
detA=1*detA
detA=detA

SECOND QUESTION:

1. The problem statement, all variables and given/known data

Let A be an nxn matrix. Prove that if matrix A satisfies 7A2+8A+3I=[0]

2. Relevant equations

Invertible Matrix Theorem

3. The attempt at a solution

7A2+8A+3I=[0]
7A2+8A = -3I
A(7A+8)=-3I

From this point I dont know if im headed towards the correct answer or have the right idea

THIRD QUESTION:

1. The problem statement, all variables and given/known data
Suppose A is an nxn matrix satisfying AT+A=[0], where n is odd. Prove detA=0.

2. Relevant equations
detAT=detA

3. The attempt at a solution

AT+A=[0]
AT=-A
detAT=det(-A)
since detAT=detA
detA=det(-A)

I think i've got the right idea...

Last edited: May 9, 2013
2. May 9, 2013

### Staff: Mentor

You can't do this (above). Matrix multiplication is not generally commutative, so in general, AP ≠ PA
Typo?
How does this show that det(A) = 0?
Also, what role does the fact that n is odd play?

3. May 9, 2013

### muzziMsyed21

sorry for the typos I edited and fixed them

4. May 9, 2013

### muzziMsyed21

that is where i am stumped... is it det(A) = -det(A) then detA=0 ?

5. May 9, 2013

### Dick

IF det(A)=(-det(A)) then det(A)=0, sure. det(-A) isn't always equal to -det(A). What is det(kA) for k a constant?

6. May 9, 2013

kndet(A)

7. May 9, 2013

### Dick

Well, sure. Now put k=(-1) and figure out what n being odd might have to do with it.

8. May 9, 2013

### muzziMsyed21

det(AT)=det(-A)
det(A)=(-1)ndetA

from here i still don't know how det=0

9. May 9, 2013

### Dick

It only has to be zero if n is odd. Then (-1)^n=(-1), right? det is a real number. If a real number x=(-x) then x=0. Can you prove that?

10. May 9, 2013

### muzziMsyed21

does it have anything to do with additive inverse?

11. May 9, 2013

### Staff: Mentor

Don't overthink this.

Since n is odd, (-1)n = -1, so
det(AT) = det(-A) = (-1)ndet(A) = ?
What can you conclude from the above?

12. May 9, 2013

### Staff: Mentor

Also, it's better to post one problem per thread.

13. May 9, 2013

### Staff: Mentor

The last line should be A(7A + 8I) = -3I
It doesn't make sense to add a number to a matrix.

Rearranging a bit results in this equation.
So A * (-1/3)(7A + 8I) = I

Does this give you any ideas?