A value of $b$ for which the rank of the matrix $A=\left[\begin{array}{cccc}1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1…

A value of $b$ for which the rank of the matrix $A=\left[\begin{array}{cccc}1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3\end{array}\right]$ is 3 , is
  1. -2
  2. -4
  3. -6
  4. 3

Solution

Given, $ A=\left[\begin{array}{cccc} 1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{array}\right] $ For rank to be 3, there must exist 3 non zero row. Now, applying $R_2 \rightarrow R_2-4 R_1 ; R_3 \rightarrow R_3-2 R_1$ $ =\left[\begin{array}{cccc} 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & 1 \\ b-2 & 0 & 4 & 2 \\ 9 & 9 & b & 3 \end{array}\right] $ Applying $R_4 \rightarrow R_4-9 R_1$ $ =\left[\begin{array}{cccc} 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & 1 \\ b-2 & 0 & 4 & 2 \\ 0 & 0 & b+9 & 3 \end{array}\right] $ Again, applying $R_4 \rightarrow R_4-3 R_2$ $ A=\left[\begin{array}{cccc} 1 & 1 & -1 & 0 \\ 0 & 0 & 1 & 1 \\ b-2 & 0 & 4 & 2 \\ 0 & 0 & b+6 & 0 \end{array}\right] $ If rank $=3$, then Last row must have all elements 0 . $ \therefore \quad b+6=0 \Rightarrow b=-6 $

Asked in: AP EAMCET 2019 (21 Apr Shift 1)

Practice more Matrices questions on Aicharya