Question

If the elements of the row of the matrix are in proportion with the elements of all other rows of the matrix, then the rank of the matrix is ________
A. 1
B. 2
C. 3
D. 4

Answer 1

For answering this question let us assume a matrix of order $3\times 3$ . For example
A = $\left[ \begin{matrix} x & y & z \\\ ax & ay & az \\\ bx & by & bz \\\ \end{matrix} \right]$ . We will find the rank of the matrix by performing various row operations and trying to make them zero as we know that the matrix that we will get after performing row operations will be equivalent to the initial one. As we know that the rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix.

Complete step by step answer:
Let us understand the terms first. The rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix. Let us assume a matrix of order $3\times 3$,
A = $\left[ \begin{matrix} x & y & z \\\ ax & ay & az \\\ bx & by & bz \\\ \end{matrix} \right]$ .
We will find the rank of the matrix by performing various row operations and trying to make them zero as we know that the matrix that we will get after performing row operations will be equivalent to the initial one.
Let us apply the following row operations.
$\begin{aligned} & {{R}_{2}}\to {{R}_{2}}-a{{R}_{1}} \\\ & {{R}_{3}}\to {{R}_{3}}-b{{R}_{1}} \\\ \end{aligned}$
After performing them the resultant equivalent matrix will be
A = $\left[ \begin{matrix} x & y & z \\\ 0 & 0 & 0 \\\ 0 & 0 & 0 \\\ \end{matrix} \right]$
As we know that rank of matrix = number of non-zero rows of the matrix.
Therefore, we get the rank of A = 1

So, the correct answer is “Option A”.

Note: We should be careful with the row operations we choose. They make a lot of difference in the answer. We should choose the one that simplifies our matrix. Here we applied row transformations since each row had elements in terms of x, y, z and we could perform operations. We must note that rank is the number of non-zero rows of a matrix and must not get confused with the order of the matrix.