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.