JEE Advanced2023MathematicsMatricesActual
Let M = a i j , i , j ∈ 1 , 2 , 3 , be the 3 × 3 matrix such that a i j = 1 if j + 1 is divisible by i , otherwise a i j = 0 . Then which of the following statements is(are) true?
Options
- AM is invertible
- BThere exists a nonzero column matrix a 1 a 2 a 3 such that M a 1 a 2 a 3 = - a 1 - a 2 - a 3
- CThe set X ∈ ℝ 3 : M X = 0 ≠ 0 , where 0 = 0 0 0
- DThe matrix ( M - 2   I ) is invertible, where I is the 3 × 3 identity matrix
Correct answer
B. There exists a nonzero column matrix a 1 a 2 a 3 such that M a 1 a 2 a 3 = - a 1 - a 2 - a 3
Step-by-step solution
Given, M = a i j ,   i ,   j ∈ 1 , 2 , 3 a i j = 1  if  j + 1  is divisible by  i otherwise  a i j = 0 So, a 11 = 1 ,   a 12 = 1 ,   a 13 = 1 ,   a 21 = 1 ,   a 22 = 0 . . . . . . , a 33 = 0 So, M = 1      1      1 1      0      1 0      1      0 ⇒ | M | = 1 ( − 1 ) − 1 ( − 1 ) = − 1