Matrices equation

[abbaba]

Hey,
I'm preparing for job interview and some of the questions will look like:

The only idea I had was to multiply D^-1 from left and E^-1 from the right but it's too easy.
I think it might be related to eigenvalues and eigenvectors.

any ideas about how to solve it?

 

[Steven G]

Multiply both sides on the left by D^-1 which basically removes the D

Now you have (X+3I)E = 5(F+E)

Now distribute the E on the lhs and the 5 on the rhs. Now solve for X.

Please try this and post your work. if it has a mistake we will guide you from there.

 

[abbaba]

What do you mean by "distribute the E" and the 5?
I thought multiply E^-1 from the right so the equation will be: (X+3I) = 5FE^-1 + I.
Maybe then I can move 3I from the lhs to the rhs (X = 5FE^-1 -4I) but how that would help?

 

[Steven G]

What you did just has a couple of mistakes. You said that 1 - 3I = -4i when it should be -2I. However you failed to note that it should have been 5(FE^-1 + I) and not 5FE^-1 + I

You asked how would it help if you arrived at X = 5FE^-1 -4I? You now know what X equals as the question asked for. You can only solve for X in terms of F and E (and their inverses).

Try to clean up those errors.

I was saying to do:

(X+3I)*E = XE + 3E

5(F+E) = 5F + 5E

So XE + 3E = 5F + 5E

Then XE = 5F + 2E

So X = (5F+2E)*E^-1 or X = 5FE^-1 + 2I

 

[abbaba]

I probably calculated it fast and wrote -4 instead of -2.
I thought this is too easy and because of that X=5FE^-1+2I maybe is not enough,
Anyway, thank you very much, I will read again my Linear algebra notebook again