Q.Classify each of the following matrices as square, null, diagonal, scalar and/or identity, giving reasons: $P=\begin{pmatrix}4&0\\0&4\end{pmatrix}$, $Q=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, $R=\begin{pmatrix}0&0\\0&0\end{pmatrix}$, $S=\begin{pmatrix}2&0\\0&5\end{pmatrix}$.

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.

• (d) $\begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{bmatrix}$ has order $3\times 4$ (3 rows, 4 columns) — not 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 singleton matrix. (Being square of order one, it is also trivially a square matrix.)

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

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

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

Options (b) and (c) are not square, and (d) has all entries $1$. Only option (a), $\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$, is square with 1's on the diagonal and 0's elsewhere.

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

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

$(1\ \ 2\ \ 3\ \ 4)$

has one row and four columns, i.e. order $1\times 4$, so it is a row matrix (it is not square, not a single column, and not a zero matrix).

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 $0$:

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

This matches option (c) exactly; the other options either reverse the rule or make all entries equal.

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 non-singular) an inverse, which is why the idea matters in the CHSE Odisha +2 Business Mathematics course.

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 $x - y = -1$:

$x - 2x = -1 \;\Rightarrow\; -x = -1 \;\Rightarrow\; x = 1, \quad y = 2x = 2.$

Hence $x = 1,\; y = 2,\; z = 4,\; w = 5$.