What is the Determinant of a 2x2 Matrix Multiplied by its Adjoint Inverse?

[Temp0]

Homework Statement

If A is a 2x2 matrix, then det (2A * adj(A)^-1) = ?

Homework Equations

Adj(A)A = det(A)I

The Attempt at a Solution

First, I separated them so it became det(2A) * det (1/ adj(A))
Then taking the 2 out, and it becomes 2^2, so 4 det(A) * det(1/ adj(A))
adj(A) = det(A) * A^-1, rearranged from the equation above.
So: 4 det(A) * det (A/det(A)), and I get stuck at around here, because I end with
4 det(A) ^2 * det (1/det(A)), however I don't know what the determinant of the determinant of A is. Could someone clarify this for me? thank you in advance.

 

[Dick]

Homework Statement

If A is a 2x2 matrix, then det (2A * adj(A)^-1) = ?

Homework Equations

Adj(A)A = det(A)I

The Attempt at a Solution

First, I separated them so it became det(2A) * det (1/ adj(A))
Then taking the 2 out, and it becomes 2^2, so 4 det(A) * det(1/ adj(A))
adj(A) = det(A) * A^-1, rearranged from the equation above.
So: 4 det(A) * det (A/det(A)), and I get stuck at around here, because I end with
4 det(A) ^2 * det (1/det(A)), however I don't know what the determinant of the determinant of A is. Could someone clarify this for me? thank you in advance.

If it's a 2x2 matrix then det(I/det(A))=1/det(A)^2, yes?

 

[slider142]

The determinant of A is just a number, like 2 was.

 

[Temp0]

Ohhhh! This was actually really helpful to another question I had... so if it was a 3x3 matrix, the det(1+2detA) where detA = 2 would become det(5), and since the matrix is 3x3, det5 becomes 5^3 = 125?

Oh one more thing, it would be the exact same as in (1+2detA)^3

 

[slider142]

... so if it was a 3x3 matrix, the det(1+2detA) where detA = 2 would become det(5), and since the matrix is 3x3, det5 becomes 5^3 = 125?

Hmm, not precisely. 1 + 2det(A) is a number, not a matrix, so its determinant would have to be itself, if we identified it as a 1x1 matrix. If you had to compute det((1 + 2det(A))*A) = det(5*A) where A is a 3x3 matrix, then you would get (5^3)*det(A), through the usual theorem on computing the determinant of a matrix multiplied by a number.

 

[Temp0]

Hmm, not precisely. 1 + 2det(A) is a number, not a matrix, so its determinant would have to be itself, if we identified it as a 1x1 matrix. If you had to compute det((1 + 2det(A))*A) = det(5*A) where A is a 3x3 matrix, then you would get (5^3)*det(A), through the usual theorem on computing the determinant of a matrix multiplied by a number.

I see, oh I wrote it wrong, I meant the question was det (((1 +2det(A))* I), would it be considered a coefficient of I in this case and become 5^3 det(I)?

 

[Dick]

I see, oh I wrote it wrong, I meant the question was det (((1 +2det(A))* I), would it be considered a coefficient of I in this case and become 5^3 det(I)?

Yes, that's it. And I'm sure you know what det(I) is.