Question

\(A\) is any matrix, defined by \(A = [a_{ij}]_{3\times 3}\), where \(a_{ij} = |i - j|\), then which of the following statements are true ?
A. \(A\) is symmetric matrix.
B. \(A\) is skew-symmetric matrix.
C. \(A\) is diagonal matrix.
D. \(A\) is non singular matrix.
Choose the correct answer from the options given below:

• A. A and B only
• B. B and C only
• C. A and D only
• D. B and D only

Hint:

Write the matrix first: it is \(\begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix}\). Check symmetry by comparing with its transpose and check the determinant.

Answer 1

The Correct Option is C

Step 1: Write out the matrix:
Put \(i, j = 1, 2, 3\) in \(a_{ij} = |i-j|\). The diagonal entries are \(|i-i| = 0\). The entries next to the diagonal are 1 and the corner entries are 2.
\[ A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix} \]

Step 2: Check statement A:
A matrix is symmetric when \(a_{ij} = a_{ji}\). Here \(|i-j| = |j-i|\) always, so \(A^T = A\). So A is TRUE.

Step 3: Check statement B:
A skew-symmetric matrix needs \(a_{ij} = -a_{ji}\). But \(a_{12} = 1\) and \(a_{21} = 1\), not \(-1\). So B is FALSE.

Step 4: Check statement C:
A diagonal matrix has zero everywhere off the diagonal. Here \(a_{12} = 1 \neq 0\). So C is FALSE.

Step 5: Check statement D:
A matrix is non singular when its determinant is not zero. Expand along the first row:
\[ |A| = 0(0-1) - 1(0-2) + 2(1-0) = 0 + 2 + 2 = 4 \] Since \(|A| = 4 \neq 0\), A is non singular. So D is TRUE.

Final Answer:
Statements A and D are true, and B and C are false. This is option 3. \[ \boxed{\text{A and D only}} \]