GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 1

Question

Let $P = [p_{ij}],$ where $1 \leq i,j \leq n$ is defined by $p_{ij} = i\times j, \forall i,j$ with $n > 2.$ The rank of $P$ is _______

Answer 1

As per the definition given for $P,$ row $k$ is $k$ multiplied by row number $1.$ i.e., all other rows are linearly dependent on row number $1.$ So, rank of $P$ is $1.$

For example, let $n = 3$

$P = \begin{bmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23} \\
p_{31} & p_{32} & p_{33} \\

\end{bmatrix}_{3 \times 3}$

$P = \begin{bmatrix}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9 \\

\end{bmatrix}_{3 \times 3}$

Now, perform the operation, $R_{2} \rightarrow R_{2} - 2R_{1}\;\text{and}\; R_{3} \rightarrow R_{3} - 3R_{1}$

$P = \begin{bmatrix}
1 & 2 & 3 \\
0 & 0 & 0 \\
0 & 0 & 0 \\

\end{bmatrix}_{3 \times 3}$

$\therefore$ The rank of the matrix is $1.$

So, the correct answer is $1.$