Q.(f) If $\begin{pmatrix} x-y & z \\ 2x-y & w \end{pmatrix} = \begin{pmatrix} -1 & 4 \\ 0 & 5 \end{pmatrix}$, then the respective values of $x, y, z, w$ are

A square matrix has equal rows and columns; option (d) is $3\times4$, so it is not square.

The order of a matrix is written as $(\text{rows}) \times (\text{columns})$. A matrix is square only if rows $=$ columns.

• (a) $[5]$ has order $1\times 1$ — square.
• (b) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ has order $2\times 2$ — square.
• (c) $\begin{bmatrix} 16 & 17 & 18 \\ 15 & 14 & 13 \\ 12 & 11 & 10 \end{bmatrix}$ has order $3\times 3$ — square. …

A matrix having only one element is a singleton matrix (order $1\times 1$).

Matrices are named by their structure: a matrix with one row is a row matrix, with one column a column matrix, and a matrix that consists of only one element is of order $1\times 1$ and is called a ** …

Square, diagonal $=1$, off-diagonal $=0$ ⇒ option (a).

An identity matrix $I$ is a square matrix in which

$a_{ij}=\begin{cases}1,& i=j\\0,& i\neq j.\end{cases}$

…

Order $1\times 4$ (a single row) ⇒ row matrix.

A matrix having only one row is called a row matrix. The given array

$(1\ \ 2\ \ 3\ \ 4)$ …

Diagonal entries $1$, off-diagonal entries $0$ ⇒ option (c).

The unit (identity) matrix is the square matrix with every principal-diagonal element equal to $1$ and all remaining elements equal to $0$:

$a_{ij}=\begin{cases}1,& i=j\\0,& i\neq j.\end{cases}$

…

The one-word term is Square matrix.

A matrix of order $m \times n$ has $m$ rows and $n$ columns. When $m = n$ (rows = columns), it is called a square matrix, e.g. a $2 \times 2$ or $3 \times 3$ matrix such as $\begin{pmatrix} 2 & 1 \\ 5 & 2 \end{pmatrix}$. Only square matrices have a determinant and (when …

Correct option: (b) $x=1,\,y=2,\,z=4,\,w=5$.

Two matrices are equal iff their corresponding elements are equal. Comparing

$\begin{pmatrix} x-y & z \\ 2x-y & w \end{pmatrix} = \begin{pmatrix} -1 & 4 \\ 0 & 5 \end{pmatrix}$

gives four equations:

$x - y = -1, \quad z = 4, \quad 2x - y = 0, \quad w = 5.$

From $2x - y = 0$ we get $y = 2x$. Substitute into $x - y = -1$: …