Question

Find the rank of the matrix $\left[ \begin{matrix} 4 & 2 & 1 & 3 \\\ 6 & 3 & 4 & 7 \\\ 2 & 1 & 0 & 1 \\\ \end{matrix} \right]$.

Answer 1

Hint: Find the number of rows and columns if $r < c$, then r is the rank or else c is the rank.
Given in the question is a $3\times 4$ matrix which is a $r\times c$ matrix.
Where r is the number of rows $\Rightarrow r=3$
c is the number of columns $\Rightarrow c=4$
Complete step-by-step answer:
The set contains four columns each having three elements.
The rank of a matrix is defined as
A) The maximum number of linearly independent column vectors in the matrix.
B) The maximum number of linearly independent row vectors in the matrix.
For a $r\times c$ matrix,
A) If $r < c$, then the maximum rank of the matrix is ‘r’.
B) If $r > c$, then the maximum rank of the matrix is ‘c’.
The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be 1.
Here, $r\times c=3\times 4$
Here, $r < c$ i.e., 3<4.
$\therefore$ Rank of matrix = 3

Note: The rank of a matrix can be found by comparing the number of rows and number of columns.
For a matrix containing the same number of rows and columns, find the determinant for the same to find the rank.