Question

Let three matrices given as ${\text{A = }}\left[ \begin{gathered} {\text{ - 1 - 2 - 3}} \\\ {\text{ 3 4 5}} \\\ {\text{ 4 5 6}} \\\ \end{gathered} \right]{\text{ , B = }}\left[ \begin{gathered} 1{\text{ - 2}} \\\ {\text{ - 1 2}} \\\ \end{gathered} \right]\;{\text{and C = }}\left[ \begin{gathered} 2{\text{ 0 0}} \\\ {\text{0 2 0}} \\\ {\text{0 0 2}} \\\ \end{gathered} \right]{\text{ }}{\text{. }}$ If a, b and c respectively denote the ranks of A, B and C then the correct order of these numbers is
A) a < b < c
B) c < b < a
C) b < a < c
D) a < c < b

Answer 1

Hint: In this question we must know that Rank of Matrix = Number of linearly independent rows or columns. Moreover, to solve this type of problems one must have the basic knowledge of matrices.

Complete step-by-step answer:

2{\text{ 0 0}} \\\ {\text{0 2 0}} \\\ {\text{0 0 2}} \\\ \end{gathered} \right]$$ It has three linearly independent rows. Hence Rank of C = 3. $${\text{B = }}\left[ \begin{gathered} 1{\text{ - 2}} \\\ {\text{ - 1 2}} \\\ \end{gathered} \right]$$In this we can see that row 2 is a scalar multiple of row 1 using -1. Here the ${2}^{nd}$ row is dependent on the ${1}^{st}$ row whereas, ${1}^{st}$ row is independent. Hence Rank of B = 1 (Because it has only one linearly independent row) $${\text{A = }}\left[ \begin{gathered} {\text{ - 1 - 2 - 3}} \\\ {\text{ 3 4 5}} \\\ {\text{ 4 5 6}} \\\ \end{gathered} \right]$$ In this the last row can be derived from the ${2}^{nd}$ row by adding 1. Hence the rank of A = 2 (Because it has two linearly independent rows) Hence b < a < c is the right answer which is option C. Note: Here we decide the ranks of matrices, by knowing how many rows or columns are dependent. For example, if one row or column depends on another in $${\text{3}} \times {\text{3}}$$ matrix then it’s rank will be 2 because it will be having 2 linearly independent rows or columns.