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UPSC 2015 Maths Optional Paper 1 Q1b — Step-by-Step Solution

10 marks · Section A

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

Question

Reduce the following matrix to row echelon form and hence find its rank:

Technique

Gaussian elimination producing row echelon form; count non-zero rows.

Solution

Let $A$ denote the given matrix. Apply elementary row operations.

Step 1 — Eliminate column 1 below the pivot

$R_2\to R_2-2R_1,\;R_3\to R_3-R_1,\;R_4\to R_4-8R_1$:

Step 2 — Eliminate column 2 below the second pivot

Pivot in $R_2$ is $-3$. Use $R_3\to R_3+R_2$ and $R_4\to R_4-5R_2$:

This is row echelon form.

Step 3 — Read off rank

There are 2 non-zero rows. Hence

Answer