Quandaries and Queries
Hi, This is Sue (again) seeking more help preparing for the MA math Teachers exam.
Level is secondary to college.
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.
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.
First of all your answer to the second question is correct.
Your first question is to solve for X if
I would first add -2 B A - I to both sides of the equation to get
Now multiply both sides of the equation, on the left, by B-1. This gives
Using the distributive law on the right side of the equation you get
But B-1 B = I so
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.