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**Hint:**For solving these problems, we need to have a clear understanding of matrices and what are the type of matrices. If we clearly have a knowledge about the different types of matrices, then we can easily know which type of matrix A is $A=\left[ 1 \right]$ .

**Complete step by step answer:**

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. For example, if the dimension of the matrix is \[2\times 3\] (one should read it as "two by three"), then it is concluded that there are two rows and three columns.

The individual items in an \[m\times n\] matrix A, often denoted by ${{a}_{ij}}$ , where i and j usually vary from $1$ to m and n, respectively, are called its elements or entries. For conveniently expressing an element of the results of matrix operations, the indices of the element are often attached to the parenthesized or bracketed matrix expression (for example, ${{\left( AB \right)}_{ij}}$ refers to an element of a matrix product). In the context of abstract index notation, this ambiguously refers also to the whole matrix product.

There are various types of matrices. If all entries of A below the main diagonal are zero, A is called an upper triangular matrix. Similarly, if all entries of A above the main diagonal are zero, A is called a lower triangular matrix. If all entries outside the main diagonal are zero, A is called a diagonal matrix. The identity matrix ${{I}_{n}}$ of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to $1$ and all other elements are equal to $0$ .

Hence in the above problem, $A=\left[ 1 \right]$ signifies that there is only one element in it. It is a \[1\times 1\] matrix, hence we can say 1 is the diagonal element. Since the diagonal element is $1$ , we can say that A is an identity matrix. It is also a square matrix due to the fact that the number of rows and columns are equal in number.

**Note:**These types of problems are pretty easy to solve, but one needs to be careful otherwise small misjudgements can lead to a totally different answer. We must not get confused between the types of matrix and choose what kind of the type is by checking the number of rows and columns and also the elements of A wisely.

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