Matrices and Determinants

Section: Mathematics

Question : 7 of 56

Marks: +1, -0

Let M=(aij),i,j∈{1,2,3}, be the 3×3 matrix such that aij=1 if j+1 is divisible by i, otherwise aij=0. Then which of the following statements is(are) true?

[JEE Adv 2023 P2]

M is invertible
There exists a nonzero column matrix (
a1
a2
a3
) such that M(
a1
a2
a3
)=(
−a1
−a2
−a3
)
The set {X∈ℝ3:MX=0}≠{0}, where 0=(
0
0
0
)
The matrix (M−2l) is invertible, where I is the 3×3 identity matrix

Validate

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1	1	1	‌
1	0	1
0	1	0

−1	1	1
1	−2	1
0	1	−2