UPSC 2014 Maths Optional Paper 1 Q1b — Step-by-Step Solution

10 marks · Section A

Rank of a matrix · Linear Algebra · asked 7× in 14 yrs · Read the full method →

Question

Using elementary row or column operations, find the rank of the matrix

\begin{bmatrix}0 & 1 & -3 & -1\\ 0 & 0 & 1 & 1\\ 3 & 1 & 0 & 2\\ 1 & 1 & -2 & 0\end{bmatrix}.

Standard row reduction to echelon form.

Strategy. Reduce to row echelon form via elementary row operations; rank = number of non-zero rows.

$R_1\leftrightarrow R_4$ :

\begin{bmatrix}1 & 1 & -2 & 0\\ 0 & 0 & 1 & 1\\ 3 & 1 & 0 & 2\\ 0 & 1 & -3 & -1\end{bmatrix}

$R_3\to R_3-3R_1$ :

\begin{bmatrix}1 & 1 & -2 & 0\\ 0 & 0 & 1 & 1\\ 0 & -2 & 6 & 2\\ 0 & 1 & -3 & -1\end{bmatrix}

$R_2\leftrightarrow R_4$ :

\begin{bmatrix}1 & 1 & -2 & 0\\ 0 & 1 & -3 & -1\\ 0 & -2 & 6 & 2\\ 0 & 0 & 1 & 1\end{bmatrix}

$R_3\to R_3+2R_2$ :

\begin{bmatrix}1 & 1 & -2 & 0\\ 0 & 1 & -3 & -1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 1\end{bmatrix}

$R_3\leftrightarrow R_4$ :

\begin{bmatrix}1 & 1 & -2 & 0\\ 0 & 1 & -3 & -1\\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 0\end{bmatrix}

Row echelon form with three non-zero rows.

\boxed{\;\operatorname{rank}(A)=3.\;}