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 . BA1XB1 + 2BA + In = 0 (the symbol "0" here denotes the matrix of all 0's in it) Also, A1 or B1 is indicating inverse and "In" = for example, A1 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. Thanks, Sue 

Hi Sue, 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. Penny 

