Match List-I with List-II

List-I	List-II
Type of matrix	Conditions
(A) Square matrix A	(I) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = k & , i = j \end{cases}$, where $k \neq 0$ is constant.
(B) Scalar Matrix A	(II) $A = [a_{ij}]_{m \times m}$
(C) Diagonal matrix A	(III) $A = [a_{ij}]_{m \times m}$ where $\begin{cases} a_{ij} = 0 & , i \neq j \\ a_{ij} = 1 & , i = j \end{cases}$
(D) Identity matrix A	(IV) $A = [a_{ij}]_{m \times m}$ where $a_{ij} = 0$, $i \neq j$

Choose the correct answer from the options given below:

Each type of matrix builds on the previous one with one extra condition — the simplest definition is "square matrix" and the most specific is "identity matrix."

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(A) Square Matrix → (II)

A square matrix means number of rows = number of columns (both are $m$), with no condition on the elements.

(II) just says $A = [a_{ij}]_{m \times m}$

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(C) Diagonal Matrix → (IV)

A diagonal matrix is a square matrix where every element outside the main diagonal is zero. The diagonal elements can be anything.

(IV) only requires $a_{ij} = 0$ when $i \neq j$

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(B) Scalar Matrix → (I)

A scalar matrix is a diagonal matrix where all diagonal elements are the same constant $k \neq 0$.

For example: $\begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}$ is a scalar matrix with $k = 5$

(I) requires off-diagonal $= 0$ and all diagonal entries $= k$

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(D) Identity Matrix → (III)

An identity matrix is a scalar matrix where $k = 1$.

For example: $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

(III) requires off-diagonal $= 0$ and diagonal $= 1$

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List-I	List-II
(A) Square Matrix	(II)
(B) Scalar Matrix	(I)
(C) Diagonal Matrix	(IV)
(D) Identity Matrix	(III)

The hierarchy of these matrices:

$\text{Square} \supset \text{Diagonal} \supset \text{Scalar} \supset \text{Identity}$

Therefore, the correct answer is Option 4.