Question

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?

• A. M is invertible
• B. There exists a nonzero column matrix$\begin{pmatrix}a_1\\\a_2\\\a_3\end{pmatrix}$ such that M$\begin{pmatrix}a_1\\\a_2\\\a_3\end{pmatrix}$=$\begin{pmatrix}-a_1\\\\-a_2\\\\-a_3\end{pmatrix}$
• C. The set {${x\in R^3:MX=0}$} $\neq$ 0, where 0=$\begin{pmatrix}0\\\0\\\0\end{pmatrix}$
• D. The matrix ( M-2I) is invertible, where I is the 3×3 identity matrix

Answer 1

Answer: (B), (C)

There exists a nonzero column matrix$\begin{pmatrix}a_1\\\a_2\\\a_3\end{pmatrix}$ such that M$\begin{pmatrix}a_1\\\a_2\\\a_3\end{pmatrix}$=$\begin{pmatrix}-a_1\\\\-a_2\\\\-a_3\end{pmatrix}$

Answer 2

Given :
M = (aij), i, j ∈ {1, 2, 3},
aij = 1 if j + 1 is divisible by i, otherwise aij = 0
$M=\begin{bmatrix} 1 & 1 & 1 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{bmatrix}$
|M| = 1(-1) - 1(-1)
= -1 + 1 = 0
So, M is not invertible
$\begin{bmatrix} 1 & 1 & 1 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} a_{1} \\\ a_{2} \\\ a_{3} \end{bmatrix}=\begin{bmatrix} -a_{1} \\\ -a_{2} \\\ -a_{3} \end{bmatrix}$
$\begin{bmatrix} a_1+ a_2 +a_3 \\\ a_1+a_3 \\\ a_2 \end{bmatrix}=\begin{bmatrix} -a_{1} \\\ -a_{2} \\\ -a_{3} \end{bmatrix}$

There exists an infinite number of possible column matrices.
$\begin{bmatrix} 1 & 1 & 1 \\\ 1 & 0 & 1 \\\ 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} x \\\ y \\\ z \end{bmatrix}=\begin{bmatrix} 0 \\\ 0 \\\ 0 \end{bmatrix}$
x + y + z = 0
⇒ x + z = 0
y = 0
So, this is possible only.
$|M-2I|=\begin{bmatrix} -1 & 1 & 1 \\\ 1 & -2 & 1 \\\ 0 & 1 & -2 \end{bmatrix}$
$=-1(3)-1(-2-1)=-3+3=0$
So, the correct options are (B) and (C).