Question

If the matrix $A = $ has rank 3, then

• A. $y \ne (a+b+c)$
• B. $y \ne 1$
• C. $y = 0$
• D. $y \ne -(a+b+c)$ and $y \ne 0$

Hint:

If the matrix $A = $ has rank 3, then

Answer 1

The Correct Option is D

Step 1: Concept
A $3 \times 3$ matrix has rank 3 if and only if its determinant is non-zero.
Step 2: Analysis
Calculate $\det(A)$. Applying $C_1 \to C_1 + C_2 + C_3$ and factoring out $(y+a+b+c)$, we get $(y+a+b+c) \begin{vmatrix} 1 & b & c \\ 1 & y+b & c \\ 1 & b & y+c \end{vmatrix}$.
Step 3: Evaluation
Simplifying the determinant using row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$, we get $(y+a+b+c)y^2 \ne 0$.
Step 4: Conclusion
Thus, $y \ne 0$ and $y \ne -(a+b+c)$.
Final Answer: (d)