Activity 2.2.2

Find the matrix product $$\left[ \begin{array}{rrrr} 1 & 2 & 0 & -1 \\ 2 & 4 & -3 & -2 \\ -1 & -2 & 6 & 1 \\ \end{array} \right] \left[ \begin{array}{r} 3 \\ 1 \\ -1 \\ 1 \\ \end{array} \right]$$.

Suppose that $A$ is the matrix $$\left[ \begin{array}{rrr} 3 & -1 & 0 \\ 0 & -2 & 4 \\ 2 & 1 & 5 \\ 1 & 0 & 3 \\ \end{array} \right]$$. If $A\xvec$ is defined, what is the dimension of the vector $\xvec$ and what is the dimension of $A\xvec$?

A vector whose entries are all zero is denoted by $\zerovec$. If $A$ is a matrix, what is the product $A\zerovec$?

Suppose that $I = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$ is the identity matrix and $\xvec=\threevec{x_1}{x_2}{x_3}$. Find the product $I\xvec$ and explain why $I$ is called the identity matrix.

Suppose we write the matrix $A$ in terms of its columns as $$A = \left[ \begin{array}{rrrr} \vvec_1 & \vvec_2 & \cdots & \vvec_n \\ \end{array} \right]$$. If the vector $\evec_1 = \left[\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right]$, what is the product $A\evec_1$?

Suppose that $$A = \left[ \begin{array}{rrrr} 1 & 2 \\ -1 & 1 \\ \end{array} \right], \bvec = \left[ \begin{array}{r} 6 \\ 0 \end{array} \right]$$. Is there a vector $\xvec$ such that $A\xvec = \bvec$?