Q.A = $[a_{ij}]_{m \times n}$ is a square matrix, if (A) $m < n$ (B) $m > n$ (C) $m = n$ (D) None of these

The key idea is the definition of a square matrix: a matrix where the number of rows equals the number of columns.

• A matrix $A = [a_{ij}]_{m \times n}$ has $m$ rows and $n$ columns.
• For it to be square, the order must satisfy $m = n$. …

The definition of a square matrix is one of the first things you learn in matrix algebra, and it’s beautifully simple. A matrix is just a rectangular arrangement of numbers, and its size is given by the number of rows ($m$) and columns ($n$). The word “square” comes from geometry — a square has equal sides. Similarly, a square matrix has equal dimensions: the row count equals the column count.

Let’s walk through the options:

1. Option (A): $m < n$

   If the number of rows is less than the number of columns, the matrix is “tall” or “portrait” shaped — it’s rectangular, not square. For example, a $2 \times 3$ matrix has 2 rows and 3 columns. This is not square.

2. Option (B): $m > n$

   Here, rows outnumber columns — a “wide” or “landscape” rectangle. A $4 \times 2$ matrix is rectangular, not square.

3. Option (C): $m = n$

   This is the only case where the matrix has the same number of rows and columns. For instance, a $3 \times 3$ matrix is square. The notation $[a_{ij}]_{n \times n}$ is often used for square matrices, where $i$ and $j$ both run from 1 to $n$.

4. Option (D): None of these …

Method: Identifying a Square Matrix From Its Order

Use this whenever a question asks you to classify a matrix by shape from its order $[a_{ij}]_{m\times n}$.

Steps

Step 1: Read off the order convention.

In $[a_{ij}]_{m\times n}$, $m$ is the number of rows and $n$ the number of columns — always rows first, then columns.

Step 2: Apply the defining condition of the type.

A matrix is square exactly when the number of rows equals the number of columns, i.e. $m=n$. If $m<n$ or $m>n$ the matrix is rectangular. …