# True / False

1. Apr 20, 2010

### System

Hi

1) Every linear system consisting of 3 equations in 4 unknowns has infintely 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 :/

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

2. Apr 20, 2010

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.

3. Apr 20, 2010

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

4. Apr 21, 2010