# Need Explanations For True/False Questions

• Oct 26th 2006, 11:11 AM
TreeMoney
Need Explanations For True/False Questions
I know the following answers are correct, I just need to know why they are correct. If someone could give me a brief explanation as to why. It will greatly help me in understand them. TIA!!!!

1. The equation A(nxm)X = b has a unique solution for all b contained in R^n if and only if the rank of A is n. (A is an n x m matrix.) False. But why?

2. Ax = 0 has more than one solution if and only if the columns fo A form a linearly depenedent set. True, but why?

3. If Ax = b has a general solution u + x2v + x4w then AB = 0 where B is the matrix with columns v and w. (x2 and x4 are x sub 2 and x sub 4) True, but why??

4. If A and B are invertible n x n matrices, then A + B is invertible. False, but why?????
• Oct 26th 2006, 04:30 PM
ThePerfectHacker
Quote:

Originally Posted by TreeMoney

4. If A and B are invertible n x n matrices, then A + B is invertible. False, but why?????

Consider,
$\displaystyle A=\left[ \begin{array}{cc}1&0\\0&1 \end{array} \right]$
$\displaystyle B=\left[ \begin{array}{cc}-1&0\\0&1 \end{array} \right]$
Note that,
$\displaystyle \det (A), \det (B)\not = 0$
Thus, $\displaystyle A,B$ are intertible.
But,
$\displaystyle A+B=\left[ \begin{array}{cc}0&0\\0&0 \end{array} \right]$
With,
$\displaystyle \det (A+B)=0$
Thus it is not invertible.