Solving Linear Systems: True or False?

[System]

Hi

Answer as T or F:

1) Every linear system consisting of 3 equations in 4 unknowns has infinitely many solutions.
2) If A and B are 3 x 3 matrices , then det(AB - A (B^T) ) = 0
3) If A and B are n x n matrices, nonsingular matrices and AB=BA, then A(B^-1) = (B^-1)A
4) If A is a singular n x n matrix, then Aadj(A)=0

For (1):
I think its true
since # of columns > # of rows
so we will have recall a parameter
and this means we will a infinitely many solutions

For (2):
I do not know how to do it =(

For (3):
I got the answer, its true
but how ?

For (4):
I completely stopped here :/

Any help please?


this is not for my homework
I swear
am solving these for fun

 

[radou]

A hint for the second one: factor out A and use the rule det(XY)=det(X)det(Y). Then, can you conclude something about det(B - B^T)? What are the diagonal elements of that matrix? What is the general element of that matrix? Use the rule of Sarrus to calculate the determinant.

 

[radou]

A hint for the third one: multiply the left side of the equation AB = BA by B^-1.

 

[radou]

A hint for the fourth one: take an element cij from the matrix C = Aadj(A) and write down what it equals. Can you conclude something from that?

 

[Newtime]

Correct me if I'm wrong, but for (1), I believe the answer is False. While you are correct most of the time, you have to consider the situation when the system might have 0 solutions. In general - if r is the rank of the matrix - r < n and r < m implies the system will have 0 or an infinite amount of solutions. Thus, not every system described will have an infinite amount of solutions.