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TS EAMCET 04 May 2019 Shift 2 Solved Paper

Section: Mathematics

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Question : 5 of 160

Marks: +1, -0

The rank of the matrix

\begin{bmatrix} 3 & 5 & -1 & 4 \\ 2 & 1 & 3 & -2 \\ 8 & 11 & 1 & 6 \\ -7 & -14 & 6 & -14 \end{bmatrix}

is

1
2
3
4

Solution:

Consider the matrix.

\begin{bmatrix} 3 & 5 & -1 & 4 \\ 2 & 1 & 3 & -2 \\ 8 & 11 & 1 & 6 \\ -7 & -14 & 6 & -14 \end{bmatrix}

Apply operation

R_{1} \rightarrow R_{1}-R_{2}

\begin{bmatrix} 1 & 4 & -4 & 6 \\ 2 & 1 & 3 & -2 \\ 8 & 11 & 1 & 6 \\ -7 & -14 & 6 & -14 \end{bmatrix}

Apply operation

R_{2} \rightarrow R_{2}-2 R_{1}, R_{3} \rightarrow R_{3}-8 R_{1}

and

{R_{4} \rightarrow R_{4}+7 R_{1}}

{\begin{bmatrix} 1 & 4 & -4 & 6 \\ 0 & -7 & 11 & -14 \\ 0 & -21 & 33 & -42 \\ 0 & 14 & -22 & 28 \end{bmatrix}}

Apply operation

R_{3} \rightarrow R_{3}-3 R_{2}

\begin{bmatrix} 1 & 4 & -4 & 6 \\ 0 & -7 & 11 & -14 \\ 0 & 0 & 0 & 0 \\ 0 & 14 & -22 & 28 \end{bmatrix}

Apply operation

R_{3} \leftrightarrow R_{4}

\begin{bmatrix} 1 & 4 & -4 & 6 \\ 0 & -7 & 11 & -14 \\ 0 & 14 & -22 & 28 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Apply operation

R_{3} \rightarrow R_{3}+2 R_{2}

\begin{bmatrix} 1 & 4 & -4 & 6 \\ 0 & -7 & 11 & -14 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}

Therefore, the rank of the matrix is 2

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