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UPSC 2017 Maths Optional Paper 2 Q5b — Step-by-Step Solution

10 marks · Section B

Gauss-Jordan method · Numerical Analysis · asked 2× in 13 yrs · Read the full method →

Question

Explain the main steps of the Gauss–Jordan method and apply this method to find the inverse of the matrix $\begin{bmatrix}2 & 6 & 6\\ 2 & 8 & 6\\ 2 & 6 & 8\end{bmatrix}$.

Technique

Gauss–Jordan elimination on $[A\mid I]$; reduce left block to $I$, right block becomes $A^{-1}$.

Solution

Main steps of Gauss–Jordan inversion.

1. Form the augmented matrix $[A\mid I]$ by appending the identity $I$ to $A$.
2. Apply elementary row operations to reduce the left block $A$ to the identity $I$ (forward elimination to upper-triangular, then back-elimination, with each pivot normalised to $1$).
3. Whatever row operations turn $A$ into $I$ turn $I$ into $A^{-1}$. When the left block is $I$, the right block is $A^{-1}$, i.e. $[A\mid I]\to[I\mid A^{-1}]$.

Step 1 — Augment

Step 2 — Create zeros below the first pivot

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

(The matrix is already nearly diagonal — the off-diagonals in rows 2,3 cleared at once.)

Step 3 — Eliminate the entries in row 1 above the pivots

Column 2: $R_1\to R_1-3R_2$ (since the $(1,2)$ entry is $6=3\cdot2$):

Column 3: $R_1\to R_1-3R_3$ (the $(1,3)$ entry is $6=3\cdot2$):

Step 4 — Normalise pivots ($\times\tfrac12$ each row)

Step 5 — Read off the inverse

Answer