Let $A$ be a matrix of size $3 \times 4$ such that its first two rows are $(1,1,1,1)$ and $(1,2,3,4)$, and with no zeros in the third row. In each of the following sections, provide an example of matrix $A$ that satisfies the requested condition, justifying it appropriately:\ a) (0.5 points) The third row of $A$ is a linear combination of the first two.\ b) (0.5 points) The three rows of $A$ are linearly independent.\ c) (0.5 points) $A$ is the augmented matrix of a compatible determined system.\ d) (0.5 points) $A$ is the augmented matrix of a compatible indeterminate system.\ e) (0.5 points) $A$ is the augmented matrix of an incompatible system.
Let $A$ be a matrix of size $3 \times 4$ such that its first two rows are $(1,1,1,1)$ and $(1,2,3,4)$, and with no zeros in the third row. In each of the following sections, provide an example of matrix $A$ that satisfies the requested condition, justifying it appropriately:\
a) (0.5 points) The third row of $A$ is a linear combination of the first two.\
b) (0.5 points) The three rows of $A$ are linearly independent.\
c) (0.5 points) $A$ is the augmented matrix of a compatible determined system.\
d) (0.5 points) $A$ is the augmented matrix of a compatible indeterminate system.\
e) (0.5 points) $A$ is the augmented matrix of an incompatible system.