Question

A. \(\begin{bmatrix}1& 2& 3   \\[0.3em]2 & 4 & 5   \\[0.3em]3 & 5&6   \\[0.3em] \end{bmatrix}\) is a Symmetric matrix. 
B. \(\begin{bmatrix}0 &0 &0   \\[0.3em]0& 0&0   \\[0.3em] \end{bmatrix}\) is a Null matrix.
C. \(\begin{bmatrix}1& 0& 0   \\[0.3em]0 & 2 & 0\\[0.3em]0 & 0&3\\[0.3em] \end{bmatrix}\) is an Identity matrix.
D. \(\begin{bmatrix}0& 1&2   \\[0.3em]-1 & 0 & 3   \\[0.3em]-2 & 3&0\\[0.3em] \end{bmatrix}\) is a Skew symmetric matrix.
E. \(\begin{bmatrix}\sqrt{3} &0& 0\\[0.3em]0 & \sqrt{3} & 0   \\[0.3em]0 & 0&\sqrt{3}   \\[0.3em] \end{bmatrix}\) is a Scalar matrix
Choose the correct answer from the options given below:

• A. A, B, E only
• B. B, C only
• C. D, E only
• D. B, D, E only

Answer 1

The Correct Option is A

To identify the correct properties of the given matrices, let's analyze each statement:
A. A symmetric matrix is one that is equal to its transpose. The matrix \( \begin{bmatrix}1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix} \) is symmetric because it remains the same when transposed.
B. A null matrix is a matrix in which all elements are zero. The matrix \( \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) fits this description, making it a null matrix.
C. An identity matrix is a square matrix with ones on the diagonal and zero elsewhere. The matrix \( \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} \) is a diagonal matrix but not an identity matrix as the diagonal elements are not all ones.
D. A skew-symmetric matrix is equal to the negative of its transpose, and its diagonal elements must be zero. The matrix \( \begin{bmatrix}0 & 1 & 2 \\ -1 & 0 & 3 \\ -2 & 3 & 0 \end{bmatrix} \) is skew-symmetric because its transpose is equal to its negative.
E. A scalar matrix is a diagonal matrix where all the diagonal elements are the same. The matrix \( \begin{bmatrix}\sqrt{3} & 0 & 0 \\ 0 & \sqrt{3} & 0 \\ 0 & 0 & \sqrt{3} \end{bmatrix} \) is indeed a scalar matrix as all diagonal elements are \( \sqrt{3} \).
Based on the above analysis, the correct statements are A, B, and E.