Skill: Identify row, column, square, zero, and identity matrices

First, count the rows and columns to identify a matrix’s shape: one row means row matrix, one column means column matrix, and equal numbers of rows and columns means square matrix. Then check the entries: a zero matrix contains only zeros, while an identity matrix is square, with ones on its main diagonal and zeros elsewhere.

Worked example: Classify each matrix, then use the identity matrix to multiply $M$.

$$A=\begin{bmatrix}0&0&0\end{bmatrix}, \quad B=\begin{bmatrix}2\\0\\-1\end{bmatrix}, \quad C=\begin{bmatrix}0&0\\0&0\end{bmatrix}, \quad I=\begin{bmatrix}1&0\\0&1\end{bmatrix}, \quad M=\begin{bmatrix}3&-2\\1&4\end{bmatrix}.$$

1. Count dimensions.
   $A$ is $1\times3$, so it is a row matrix. $B$ is $3\times1$, so it is a column matrix. Both $C$ and $I$ are $2\times2$, so they are square matrices.

2. Check the entries.
   Every entry of $A$ and $C$ is zero, so both are zero matrices. The matrix $I$ has ones on its main diagonal and zeros elsewhere, so it is an identity matrix. Notice that $A$ is both a row matrix and a zero matrix: these categories describe different features.

3. Multiply by the identity.
   Since $I$ and $M$ are both $2\times2$, the product is defined:

$$IM= \begin{bmatrix}1&0\\0&1\end{bmatrix} \begin{bmatrix}3&-2\\1&4\end{bmatrix} = \begin{bmatrix}3&-2\\1&4\end{bmatrix} =M.$$

The identity matrix leaves a compatible matrix unchanged under multiplication.