Matrix Proofs Homework: Q1 & Q2

[colt]

Homework Statement

First Question: Be A and B square matrix. Show that if A and B are invertible matrix, then: (A + B)^(-1) = A^(-1) * [I + B*A^(-1)]^(-1)

The Attempt at a Solution

First Queston: (A+B)^(-1) = [A^(-1) + A^(-1)*B*A^(-1)]^(-1) (I distributed the a^(-1) outside the []). Then I don't know what else I can do

Homework Statement

Second Question: Be A and P matrix of nxn dimension with P being invertible. So det (P^(-1)*A*P) = det(A)

The Attempt at a Solution

I would like to transform the (P^(-1)*A*P) into (I*A), but since matrix multiplication ain't commutative, I don't know how to get there

 

[jbunniii]

First Queston: (A+B)^(-1) = [A^(-1) + A^(-1)*B*A^(-1)]^(-1) (I distributed the a^(-1) outside the []). Then I don't know what else I can do

You can verify this by showing that
$$(A+B)(A^{-1}[I + BA^{-1}]^{-1}) = I$$

Second Question: Be A and P matrix of nxn dimension with P being invertible. So det (P^(-1)*A*P) = det(A)

What properties of determinants do you know about?

 

[colt]

I was only able to develop it until here:

A(A^{-1}[I + BA^{-1}]^{-1}) + B(A^{-1}[I + BA^{-1}]^{-1}) =

AA^{-1}[I + BA^{-1}]^{-}) + BA^{-1}[I + BA^{-1}]^{-1} =

I[I + BA^{-1}]^{-1} + BA^{-1}[I + BA^{-1}]^{-1} =

[I + BA^{-1}]^{-1} + B[A^{-1} + A^{-1}BA^{-1}]^{-1} =

What properties of determinants do you know about?

I know that de determinant of A^(-1) = 1/ det A, that det A = det A transpose, that the det A with a constant K that multiplies one of its lines is equal to K that multiplies det A and so that if this constant multiplies all lines. then det A = K^n * det A

 

[Dick]

I was only able to develop it until here:

A(A^{-1}[I + BA^{-1}]^{-1}) + B(A^{-1}[I + BA^{-1}]^{-1}) =

AA^{-1}[I + BA^{-1}]^{-}) + BA^{-1}[I + BA^{-1}]^{-1} =

I[I + BA^{-1}]^{-1} + BA^{-1}[I + BA^{-1}]^{-1} =

[I + BA^{-1}]^{-1} + B[A^{-1} + A^{-1}BA^{-1}]^{-1} =
I know that de determinant of A^(-1) = 1/ det A, that det A = det A transpose, that the det A with a constant K that multiplies one of its lines is equal to K that multiplies det A and so that if this constant multiplies all lines. then det A = K^n * det A

I think the problem is asking you to assume that the wrong things are invertible. A and B invertible doesn't show A+B is invertible. You need to assume that everything is invertible. Then for the first use that C^(-1)D^(-1)=(DC)^(-1). Figure out what to put D and C equal to.

 

[jbunniii]

I was only able to develop it until here:

A(A^{-1}[I + BA^{-1}]^{-1}) + B(A^{-1}[I + BA^{-1}]^{-1}) =

AA^{-1}[I + BA^{-1}]^{-}) + BA^{-1}[I + BA^{-1}]^{-1} =

I[I + BA^{-1}]^{-1} + BA^{-1}[I + BA^{-1}]^{-1} =

Now try using the distributive property.

 

[jbunniii]

I know that de determinant of A^(-1) = 1/ det A, that det A = det A transpose, that the det A with a constant K that multiplies one of its lines is equal to K that multiplies det A and so that if this constant multiplies all lines. then det A = K^n * det A

Another important property: $\det(AB) = \det(A)\det(B)$. Do you know this one? If so, the second problem should be easy.

 

[colt]

Now try using the distributive property.

How? The identity is irrelevant, so only the a^{-1} is left, and I already distributed it without seeing any meaningful result

Another important property:det(AB)=det(A)det(B). Do you know this one? If so, the second problem should be easy.

Ah yes, had forgotten this one.

I think the problem is asking you to assume that the wrong things are invertible. A and B invertible doesn't show A+B is invertible. You need to assume that everything is invertible. Then for the first use that C^(-1)D^(-1)=(DC)^(-1). Figure out what to put D and C equal to.

Didn't really understand. Can you try to explain it once more?

 

[Dick]

Didn't really understand. Can you try to explain it once more?

What I mean is take the inverse of both sides of (A + B)^(-1) = A^(-1) * [I + B*A^(-1)]^(-1). What you first did, distributing the A^(-1) inside of the [] is quite wrong. There's a ^(-1) on the bracket.

 

[jbunniii]

now try using the distributive property.

how? The identity is irrelevant, so only the a^{-1} is left, and i already distributed it without seeing any meaningful result

Both terms have a common factor on the right, namely $(I + BA^{-1})^{-1}$:

$$I(I+BA^{-1})^{-1} + BA^{-1}(I+BA^{-1})^{-1}$$

So factor out that common term and see what you get.

 

[colt]

What you first did, distributing the A^(-1) inside of the [] is quite wrong. There's a ^(-1) on the bracket.

Yes, I have to admit that I was suspicious about this distributive. Thanks for pointing it out.

Both terms have a common factor on the right, namely (I+BA−1)−1:

I(I+BA−1)−1+BA−1(I+BA−1)−1


So factor out that common term and see what you get.

I see it now. Thanks