Question

If \(A = [a_{ij}]_{3\times 3}\), \(a_{ij} = \begin{cases} 0, & i \neq j \\ i+j, & i = j \end{cases}\), then which of the following statements are correct ?
A. \(A\) is scalar matrix
B. \(A\) is diagonal matrix
C. \(A\) is unit matrix
D. \(A\) is symmetric matrix
Choose the correct answer from the options given below:

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

Hint:

Write the matrix as \(\operatorname{diag}(2,4,6)\). It is diagonal and symmetric, but not scalar or unit.

Answer 1

The Correct Option is C

Step 1: Write the matrix from the rule.
The rule gives zero when the row and column numbers differ, and $i+j = 2i$ on the diagonal. So the diagonal is $2,4,6$ and the matrix is $\operatorname{diag}(2,4,6)$.

Step 2: Use the nesting of matrix types.
A unit matrix is a scalar matrix, and a scalar matrix is a diagonal matrix. Our matrix has different diagonal entries. So it is diagonal, but it is neither scalar nor unit. This makes B true, A and C false.

Step 3: Test symmetry.
The transpose of a diagonal matrix has the same entries, because moving $(i,j)$ to $(j,i)$ only touches zero places. So $A^T=A$ and D is true.

Step 4: Choose the option.
The correct pair is B and D, which is option 3.

Step 5: Why the other options fail.
Option 1 has C, and C is false because the diagonal is 2, 4, 6 and not all ones. Option 2 has A and C, and both are false. Option 4 also has C. Only option 3 has exactly the two true statements B and D. Note that a matrix like $2I$ would be scalar, but $\operatorname{diag}(2,4,6)$ is not, because its entries are unequal.

Final Answer:
The correct statements are B and D. \[ \boxed{\text{Option 3}} \]