← 2021 Paper 1

UPSC 2021 Maths Optional Paper 1 Q4a-i — Step-by-Step Solution

10 marks · Section A

Rank of a matrix · Linear Algebra · asked 7× in 13 yrs · Read the full method →

Question

$A=\begin{bmatrix}1 & 3 & 2 & 4 & 1\\ 0 & 0 & 2 & 2 & 0\\ 2 & 6 & 2 & 6 & 2\\ 3 & 9 & 1 & 10 & 6\end{bmatrix}$. Reduce to RREF; find rank.

Technique

Standard Gaussian elimination to RREF; track pivot positions; count non-zero rows.

Solution

Step 1 — Use $R_1$ to clear column 1

$R_3\to R_3-2R_1$: $(0,0,-2,-2,0)$.

$R_4\to R_4-3R_1$: $(0,0,-5,-2,3)$.

Step 2 — Use $R_2$ to clear column 3

$R_2$ has pivot in column 3 (value 2). Divide $R_2$ by 2: $(0,0,1,1,0)$.

$R_3\to R_3+2R_2$: $(0,0,0,0,0)$.

$R_4\to R_4+5R_2$: $(0,0,0,3,3)$.

$R_1\to R_1-2R_2$: $(1,3,0,2,1)$.

Step 3 — Swap $R_3,R_4$; use new $R_3$ to clear column 4

After swap:

Divide $R_3$ by 3: $(0,0,0,1,1)$.

$R_2\to R_2-R_3$: $(0,0,1,0,-1)$.

$R_1\to R_1-2R_3$: $(1,3,0,0,-1)$.

This is RREF.

Step 4 — Read rank

Three non-zero rows. Rank = 3.

Answer