Worksheet 13

1. Determine the rank and nullity of \(\begin{pmatrix} 7 & 12 & 3 & -1 & 3 \\ -1 & 4 & 3 & 7 & 2 \\ -3 & 0 & 1 & 21 & 2 \\ 3 & 16 & 7 & 27 & 7 \end{pmatrix}\)

2. Suppose we can write a matrix $A$ as a product $A = BC$. Show that \(\operatorname{nullity} C \leq \operatorname{nullity} A\)

3. Let $A$ be an $m\times n$ matrix. Determine if the following statements are true or false:

   • If $x \mapsto Ax$ is an injective function, then the nullity of $A$ is $0$.
   • If $x \mapsto Ax$ is an injective function, then the rank of $A$ is $m$.
   • If $m=n$, then \(\operatorname{nullity} A = \operatorname{nullity} A^T.\)