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UPSC 2016 Maths Optional Paper 1 Q1a-i — Step-by-Step Solution

6 marks · Section A

Inverse of a matrix (adjoint and row reduction) · Linear Algebra · asked 3× in 13 yrs · Read the full method →

Question

Using elementary row operations, find the inverse of $A=\begin{bmatrix}1 & 2 & 1\\ 1 & 3 & 2\\ 1 & 0 & 1\end{bmatrix}$.

Technique

Gauss–Jordan elimination on $[A\,|\,I]$; the elementary row operations that drive the left block to $I$ build $A^{-1}$ on the right.

Solution

Step 1 — Set up the augmented matrix $[A\,|\,I]$

Step 2 — Clear column 1 below the pivot

$R_2\to R_2-R_1$ and $R_3\to R_3-R_1$:

Step 3 — Clear column 2

$R_1\to R_1-2R_2$ and $R_3\to R_3+2R_2$:

Step 4 — Normalize the third pivot and clear column 3

$R_3\to\frac12 R_3$, then $R_1\to R_1+R_3$ and $R_2\to R_2-R_3$:

The left block is $I$, so the right block is $A^{-1}$:

Answer