Determinants Matrices Question 6
Question: The rank of the matrix $ \begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1 \\ \end{bmatrix} $ is
Options:
A) 1 if $ a=6 $
B) 2 if $ a=1 $
C) 3 if $ a=2 $
D) 1 if $ a=4 $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] Let $ A= \begin{bmatrix} -1 & 2 & 5 \\ 2 & -4 & a-4 \\ 1 & -2 & a+1 \\ \end{bmatrix} \tilde{\ } \begin{bmatrix} -1 & 2 & 5 \\ 0 & 0 & a+6 \\ 0 & 0 & a+6 \\ \end{bmatrix} $
$ [R_2\to R_2+2R_1,R_3\to R_3+R_1] $ Clearly rank of A is 1 if $ a=-6 $ Also, for $ a=1, $
$ |A|= \begin{vmatrix} -1 & 2 & 5 \\ 2 & -4 & -3 \\ 1 & -2 & 2 \\ \end{vmatrix}=0 $ and $ \begin{vmatrix} 2 & 5 \\ -4 & -3 \\ \end{vmatrix}=-6+20=14\ne 0 $
$ \therefore $ rank of A is 2 if $ a=1 $
BETA
