Exercise 9

Match each type of matrix with all these descriptions that could fit, say ‘None’ if it applies: (i) can be multiplied by its transpose to make a $1\!\times\!1$ matrix, (ii) can represent a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{2}$ that is not onto, (iii) can represent an isomorphism from $\mathbb{R}^{3}$ to $\mathcal{P}^{2}$.

• a $2\!\times\!3$ matrix whose rank is $1$

• a $3\!\times\!3$ matrix that is nonsingular

• a $2\!\times\!2$ matrix that is singular

• an $n\!\times\!1$ column vector