The matrix $A = \begin{bmatrix}0 & 0 & 5\\0 & 5 & 0\\5 & 0 & 0\end{bmatrix}$ is a

(A) Diagonal matrix

(B) Scalar matrix

(C) Square matrix

(D) Symmetric matrix

Choose the correct answer from the options given below:

Given matrix $A = \begin{bmatrix}0 & 0 & 5\\0 & 5 & 0\\5 & 0 & 0\end{bmatrix}$

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A diagonal matrix has non-zero values only on the main diagonal. All other elements must be zero.

The main diagonal of $A$ contains: 0, 5, 0

However, $A$ has 5 at position (1,3) and 5 at position (3,1), which are outside the main diagonal.

Therefore, $A$ is not a diagonal matrix.

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A scalar matrix is a special diagonal matrix where all diagonal elements are equal and all other elements are zero.

Since $A$ is not a diagonal matrix, it cannot be a scalar matrix.

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A square matrix has equal number of rows and columns.

Matrix $A$ has 3 rows and 3 columns.

Therefore, $A$ is a square matrix.

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A symmetric matrix equals its own transpose. This means element at position $(i,j)$ equals element at position $(j,i)$.

Checking the elements:

• Position (1,3): value is 5, Position (3,1): value is 5
• Position (1,2): value is 0, Position (2,1): value is 0
• Position (2,3): value is 0, Position (3,2): value is 0

Therefore, $A$ is a symmetric matrix.

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Matrix $A$ is both a square matrix and a symmetric matrix.

The correct options are (C) and (D).