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: Understanding the Concept:
A diagonal matrix has zero entries off the main diagonal. A scalar matrix is a diagonal matrix with equal diagonal entries. A unit matrix has all diagonal entries equal to 1. A symmetric matrix satisfies \(a_{ij}=a_{ji}\).

Step 2: Build the matrix.
Off the diagonal, \(a_{ij}=0\). On the diagonal, \(a_{11}=1+1=2\), \(a_{22}=2+2=4\), \(a_{33}=3+3=6\).
\[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6 \end{bmatrix} \]

Step 3: Check statement A.
The diagonal entries 2, 4, 6 are not all equal. So \(A\) is not a scalar matrix. A is FALSE.

Step 4: Check statement B.
All entries off the main diagonal are zero. So \(A\) is a diagonal matrix. B is TRUE.

Step 5: Check statement C.
A unit matrix needs 1 on every diagonal place. Here the diagonal is 2, 4, 6. C is FALSE.

Step 6: Check statement D.
Every off diagonal entry is 0, so \(a_{ij}=a_{ji}\) for all \(i,j\). That means \(A^T=A\). D is TRUE.

Step 7: Match with the options.
The true statements are B and D. That is option 3.

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