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: Build the matrix from the rule:
Each entry is the distance between its row number and its column number. That gives 0 on the diagonal, 1 one step away and 2 two steps away. The matrix is $\begin{bmatrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{bmatrix}$.

Step 2: Test symmetry and skew-symmetry by reading rows and columns:
Row 1 is $(0,1,2)$ and column 1 is also $(0,1,2)$. Row 2 equals column 2 and row 3 equals column 3. So $A = A^T$ and $A$ is symmetric. A matrix that is both symmetric and skew-symmetric must be the zero matrix. $A$ is not zero, so $A$ is not skew-symmetric.

Step 3: Test the diagonal shape:
The entry in row 1, column 2 is 1, which is not zero. So $A$ is not diagonal.

Step 4: Test invertibility with row operations:
Row operations of the type "subtract one row from another" keep the determinant the same. First do $R_3 \to R_3 - R_2$, which gives $(1, 1, -1)$. Then do $R_2 \to R_2 - R_1$, which gives $(1, -1, -1)$. The matrix is now $\begin{bmatrix} 0 & 1 & 2 \\ 1 & -1 & -1 \\ 1 & 1 & -1 \end{bmatrix}$. Expand along the first row: \[ 0 - 1(-1+1) + 2(1+1) = 0 - 0 + 4 = 4 \] The value is not zero, so $A$ is non singular.

Final Answer:
Only A (symmetric) and D (non singular) hold, which is option 3. \[ \boxed{\text{A and D only}} \]