Let $B = 1232463694812$ and $b = 51015$ . Determine the rank of $B$ and state whether the system $B x = b$ is consistent.

medium MCQ 3 marks Class 12 · Mathematics · Matrices

Answer

Option (A) Rank 1, consistent

The correct answer is Option (A) Rank 1, consistent — Therefore the system is consistent.

Step-by-step solution

Observe that each row of $B$ is a multiple of the first row: $R_{2} = 2 R_{1}$ , $R_{3} = 3 R_{1}$ .
Hence only one row is linearly independent, so $rank (B) = 1$ .
Form the augmented matrix $[B ∣ b]$ ; the right‑hand side vector is also a multiple of the first entry: $5, 10, 15$ are $5 \times 1$ , $5 \times 2$ , $5 \times 3$ respectively.
Since the proportionality between rows of $B$ is preserved in the augmented matrix, there is no contradictory equation.
Therefore the system is consistent.

Common mistake: Assuming rank equals the number of rows without row reduction

Concept tested: Rank and consistency of a linear system