Condition for rank to be 2.

Question

Suppose that

$$\mbox{rank}\begin{equation} \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & 1 & 1 & 3\\ a & b & b & 1 \end{pmatrix} \end{equation} = 2$$

for some real numbers $a$ and $b$. What is the value of $b$?

• $0.3$
• $3$
• $1$
• $0.5$

I'm unable to get the condition on $b$ only.

Answer 1

For the rank to be $2$ the third row has to be a linear combination of the first two (since the first two are obviously linearly independent).

Then, if the two $b$s in the third row are equal, the coefficient of the first row must be $0$.

This leads to a value of $b$ that is not among the options, so there must be a bug in the exercise.

Answer 2

$$\begin{equation} \begin{pmatrix} 1 & 1 & 2 & 2\\ 1 & 1 & 1 & 3\\ 1/3 & 1/3 & 1/3 & 1 \end{pmatrix} \end{equation}$$

has rank $2$ thus the closest answer is $ b=0.3$