Question

The rank of the matrix \(\left[ {\begin{array}{*{20}{c}} { - 4}&1&{ - 1}\\ { - 1}&{ - 1}&{ - 1}\\ 7&{ - 3}&1 \end{array}} \right]\)

1. 1
2. 2
3. 3
4. 4

Answer 1

Correct Answer - Option 2 : 2

\(A = \left[ {\begin{array}{*{20}{c}} { - 4}&1&{ - 1}\\ { - 1}&{ - 1}&{ - 1}\\ 7&{ - 3}&1 \end{array}} \right]\)

\(A = \left[ {\begin{array}{*{20}{c}} 4&{ - 1}&1\\ 1&1&1\\ 7&{ - 3}&1 \end{array}} \right]\)

R₁ = R₁ – 4R₂

\(A = \left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{ - 3}\\ 1&1&1\\ 7&{ - 3}&1 \end{array}} \right]\)

R₃ = R₃ – 7R₂

\(A = \left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{ - 3}\\ 1&1&1\\ 0&{ - 10}&{ - 6} \end{array}} \right]\)

R₃ = R₃ – 2R₁

\(A = \left[ {\begin{array}{*{20}{c}} 0&{ - 5}&{ - 3}\\ 1&1&1\\ 0&0&0 \end{array}} \right]\)

The number of non-zero rows is 2, therefore the rank of the matrix is 2.

ρ(A) = 2