$\newcommand{\ones}{\mathbf 1}$

By convention, we consider $[0.1~0.3]$ and $\left[\begin{array}{c}0.1 \\ 0.3 \end{array} \right]$ to be the same.

• Incorrect.
• Correct! For two matrices to be equal, they must have the same dimensions.

The matrix $\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ -2 & 1 \end{array}\right]$ has dimensions $2 \times 3$.

• Incorrect.
• Correct! The dimensions are $3 \times 2$. Dimensions are always given as (number of rows) $\times$ (number of columns).

The $2,1$ entry of $\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ -2 & 1 \end{array}\right]$ is $0$.

• Correct! The $2,1$ entry is the one in the second row and first column.
• Incorrect.

The matrix $\left[\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}\right]$ is an identity matrix.

• Incorrect.
• Correct! An identity matrix has ones on the diagonal, and zeros in all off-diagonal entries.

Suppose we know that $\left[\begin{array}{c} a \\ b \\ c \end{array}\right] =e_2$, the second unit vector (or standard basis vector). Then we can conclude $a=0$.

• Correct! Since the vector on the left has dimension $3$, so must $e_2$. That means $e_2= \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right]$, and its first component is $0$.
• Incorrect.