Quandaries and Queries


Hi, This is Sue (again) seeking more help preparing for the MA math Teachers exam.
Level is secondary to college.
Question 1
  Suppose all matrices  in the equation below are square and invertible. Solve for x .
      BA-1XB-1 + 2BA + In = 0  (the symbol "0" here denotes the matrix of all 0's in it)
Also, A-1 or B-1 is indicating inverse and "In" = for example, A-1 times A
I hope you understand the above. I have to show all the steps.
Question 2
 Suppose we consider the set of all 2x2 matrices along with the operations of matrix addition and multiplication. Do they form a field? Why or why not?
I think the answer is no because under multiplication it is not commutative and not all square matrices are invertible. I not positive so I'd like some help.


Hi Sue,

First of all your answer to the second question is correct.

Your first question is to solve for X if

B A-1 X B-1 + 2 B A + I = 0

I would first add -2 B A - I to both sides of the equation to get

B A-1 X B-1 = -2 B A - I

Now multiply both sides of the equation, on the left, by B-1. This gives

B-1( B A-1 X B-1) = B-1 (-2 B A - I)

Using the distributive law on the right side of the equation you get

B-1 B A-1 X B-1 = -2 B-1 B A - B-1

But B-1 B = I so

A-1 X B-1 = -2 A - B-1

Continue by multipying both sides of the equation, on the right, by B and then on the left by A. This will leave X on the left and your answer on the right.

You can verify your solution by substituting the value of X you find into the left side of the equation you started with, and the expression should all simplify to 0.



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