Question

\(\text{The matrix }\) \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \text{ is a:}\)
(A) Scalar matrix
(B) Diagonal matrix
(C) Skew-symmetric matrix
(D) Symmetric matrix

• A. (A), (B), and (D) only
• B. (A), (B), and (C) only
• C. (A), (B), (C), and (D)
• D. (B), (C), and (D) only

Hint:

When analyzing a matrix, it’s important to understand its properties like scalar, diagonal, symmetric, and skew-symmetric. A scalar matrix has all diagonal elements equal, and a diagonal matrix has all off-diagonal elements equal to zero. A symmetric matrix satisfies \( A = A^T \), while a skew-symmetric matrix has \( A = -A^T \), and requires all diagonal elements to be zero. Keep these distinctions in mind when classifying matrices.

Answer 1

The Correct Option is A

\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\)
- The matrix above is a scalar matrix because all diagonal elements are equal and non-zero.
- It is also a diagonal matrix since all non-diagonal elements are zero.
- This matrix is symmetric because \( A = A^T \), where \( A^T \) is the transpose of \( A \).
- However, it is not a skew-symmetric matrix because a skew-symmetric matrix requires all diagonal elements to be zero.

Answer 2

The given matrix is: 

\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\)

Analysis:

• Scalar Matrix: The matrix is a scalar matrix because all diagonal elements are equal and non-zero. Specifically, the diagonal elements are all 1.
• Diagonal Matrix: It is also a diagonal matrix since all non-diagonal elements are zero. This matrix has only diagonal elements with non-zero values and all off-diagonal elements are zero.
• Symmetric Matrix: This matrix is symmetric because \( A = A^T \), meaning the matrix is equal to its transpose. In other words, the matrix is symmetric about the main diagonal.
• Not Skew-Symmetric: However, it is not a skew-symmetric matrix. A skew-symmetric matrix requires that all diagonal elements be zero, which is not the case here. The diagonal elements are 1, not 0.