Analyzing and Classifying the Given Matrix

The question asks us to classify the given matrix based on different types: Scalar matrix, Diagonal matrix, Skew-symmetric matrix, and Symmetric matrix. The given matrix is:

\[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

This is a square matrix of order \(3 \times 3\). Let's examine its properties against the definitions of the given matrix types.

Understanding Different Matrix Types

Let's define the types of matrices mentioned in the options:

• Diagonal Matrix: A square matrix \(A = [a_{ij}]\) is called a diagonal matrix if all its non-diagonal elements are zero, i.e., \(a_{ij} = 0\) for all \(i \neq j\). The diagonal elements \(a_{ii}\) can be any value.
• Scalar Matrix: A diagonal matrix in which all the diagonal elements are equal is called a scalar matrix. So, \(a_{ij} = 0\) for \(i \neq j\), and \(a_{ii} = c\) for some constant \(c\), for all \(i\).
• Symmetric Matrix: A square matrix \(A\) is said to be symmetric if its transpose \(A^T\) is equal to itself, i.e., \(A^T = A\). This means \(a_{ij} = a_{ji}\) for all \(i\) and \(j\).
• Skew-Symmetric Matrix: A square matrix \(A\) is said to be skew-symmetric if its transpose \(A^T\) is equal to the negative of the matrix, i.e., \(A^T = -A\). This means \(a_{ij} = -a_{ji}\) for all \(i\) and \(j\). For a skew-symmetric matrix, all diagonal elements must be zero, i.e., \(a_{ii} = 0\).

Analyzing the Given Matrix Properties

Let's apply these definitions to the given matrix \(A\):

\[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

• Is it a Diagonal Matrix? The non-diagonal elements are \(a_{12}=0, a_{13}=0, a_{21}=0, a_{23}=0, a_{31}=0, a_{32}=0\). All non-diagonal elements are indeed zero. So, the matrix \(A\) is a diagonal matrix. (Option B is correct)
• Is it a Scalar Matrix? The matrix \(A\) is a diagonal matrix. The diagonal elements are \(a_{11}=1, a_{22}=1, a_{33}=1\). Since all diagonal elements are equal (to 1), the matrix \(A\) is a scalar matrix. (Option A is correct)
• Is it a Symmetric Matrix? Let's find the transpose of \(A\): \[ A^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Since \(A^T = A\), the matrix \(A\) is a symmetric matrix. (Option D is correct)
• Is it a Skew-Symmetric Matrix? For \(A\) to be skew-symmetric, \(A^T = -A\). Also, all diagonal elements must be zero. The diagonal elements of \(A\) are 1, not 0. Furthermore, \(A^T = A\), and \(A \neq -A\) (since \(A\) is not the zero matrix). So, the matrix \(A\) is not a skew-symmetric matrix. (Option C is incorrect)

Summary of Classification

Based on our analysis, the given matrix is a:

• Scalar matrix (A)
• Diagonal matrix (B)
• Symmetric matrix (D)

It is not a Skew-symmetric matrix (C).

The given matrix is also known as the Identity matrix, which is a special case of a scalar matrix where the scalar is 1.

Matching with Options

We found that the matrix is (A) Scalar, (B) Diagonal, and (D) Symmetric.

Let's look at the options provided:

1. (A),(B)and (D) only
2. (A), (B)and (C) only
3. (A),(B), (C)and (D)
4. (B), (C) and (D) only

Our findings match option 1, which states that the matrix is (A), (B), and (D) only.

Revision Table: Matrix Properties

Additional Information: Hierarchy of Matrix Types

It is important to note that some matrix types are special cases of others. Understanding this hierarchy can be helpful:

• Every Identity Matrix is a Scalar Matrix.
• Every Scalar Matrix is a Diagonal Matrix.
• Diagonal matrices are a subset of square matrices.
• The Identity Matrix is also a Symmetric Matrix because \(I^T = I\).
• The only matrix that is both Symmetric and Skew-symmetric is the Zero Matrix (\(A = A^T = -A \implies 2A = 0 \implies A=0\)). The given matrix is not the zero matrix, so it cannot be both.

The given matrix is the Identity matrix \(I_3\), which fits the descriptions of Diagonal, Scalar, and Symmetric matrices, but not Skew-symmetric.