Two similiar determinant questions

[DanielJackins]

Homework Statement

QUESTION 1:

Suppose A is a 3x3 matrix such that det(A) = 1/3.

Then det[A^3((adj(A))^-1)^2)] =
and det[3A^-1(adj(A))] =

and

QUESTION 2:

Let A and B be 3x3 invertible matrices where det(A) = 2 and det(B) = -8

Then det[A^T(adj(B))] =
and det[A^-1(adj(B))] =

The Attempt at a Solution

QUESTION 1:

For the first blank I got the correct solution but I don't know if that was just by chance. I said A^3 = 1/27 * the inverse of the adj(A) squared. I found the inverse of the adj(A) to be 3. so 1/27 * 3 * 3 = 3 which was correct.

For the second blank I said 3A^-1 is equal to 81 (because I believe you have to raise 3 to the power of the dimension of the matrix?). Then inside the brackets I tried to do the adj(A), which if I'm not mistaken is equal to det(A)*A^-1. det(A) = 1/3, which I then raised to the power of 3 as well, and multiplied by A^-1, which is 3. So all in all I had 81(1/27*3) which was not correct. Any help?

QUESTION 2:

For the first blank I said 2((-8^3)*(-1/8)) using the same logic as above and arrived at 128. (This may actually be correct as I haven't checked but I only have one attempt left and I want to be sure it's correct).

The second blank I did the same but with 1/2 on the outside as opposed to 2, and arrived at 32.

Any help would be greatly appreciated

 

[mighty2000]

!

Hello fellow scientist,

For the first blank in question 1, you are correct in saying that A^3 = 1/27 * the inverse of the adj(A) squared. However, the inverse of the adj(A) is not 3, it is actually 9. So the correct answer would be 1/27 * 9 * 9 = 3.

For the second blank, you are on the right track but there are a couple of errors in your calculations. First, you are correct in saying that 3A^-1 is equal to 81. However, when finding the adj(A), you do not need to raise det(A) to the power of 3. Instead, you can simply use the value of det(A) which is 1/3. So the correct answer would be 81 * (1/3 * 3) = 27.

For the first blank in question 2, you are correct in saying that the answer is 128. This can be found by using the formula det(A^T) = det(A).

For the second blank, you have the correct idea but there is a small error in your calculation. The correct answer would be 32 * (-1/8) = -4. This is because when finding the adj(B), you need to use the value of det(B) which is -8, not 1/2.

I hope this helps and good luck with your future calculations!