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

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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.

Updated On: Mar 28, 2025
  • (A), (B), and (D) only
  • (A), (B), and (C) only
  • (A), (B), (C), and (D)
  • (B), (C), and (D) only
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The Correct Option is A

Approach Solution - 1

\(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\)
- 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.
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Approach Solution -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.
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