-1
$\begingroup$

If $A$ is a $4\times2$ matrix and $B$ is a $2\times 3$ matrix, what are the possible values of $\operatorname*{rank}(AB)$?

Construct examples of $A$ and $B$ exhibiting each possible value of $\operatorname*{rank}(AB)$ and explain your reasoning.

  • 2
    Do not just copy-paste exact questions without your attempts. It's not everyone's homework. – 2017-01-29
  • 0
    What have you tried so far? Do you know exactly how many rows and columns the AB matrix will have? – 2017-01-29
  • 0
    It will be a 4*3 Matrix but I am confused with the statement "construct examples". How to come up with all the examples that illustrate all possible values ? – 2017-01-29
  • 0
    There are not so many cases. Begin with very simple matrices $A$ and $B$ with many zeros and a few ones... and compute $AB$ and determine its rank each time... – 2017-01-29

2 Answers 2

3

Hint. Use that in general, $$\text{rank}(AB)\leq \min\{\text{rank }A, \text{rank }B\},$$ and in our case, $$0\le \text{rank }A \le 2,\quad 0\le \text{rank }B \le 2.$$

0

rank = number of pivots in reduced form

Okay, so in terms of pivots per column, matrix A can have at most 2 pivots and matrix B also 2 pivots at most. AB which is 4*3 can have 3 pivots at most so I don't really understand why the formula above is true.