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$\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 3 &12 &24 &21 \end{bmatrix} = 3\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 1 &4 &8 &7 \end{bmatrix} $

$R_1$ and $R_4$ are the same and hence we can remove $R_4$ making the rank surely less than $4$.

$\text{Taking 3 out from $R_2$} \implies 9\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 1 &0\\ 4 &2& 3 &1 \end{bmatrix} $

${R_{1} \leftarrow R_{1}-8R_{2} \atop{R_{3} \leftarrow R_{3}-3R_{2}} } \implies 9\begin{bmatrix} 1 &4 &0 &7\\ 0 &0& 1 &0\\ 4 &2& 0 &1 \end{bmatrix} $

None of the rows are linearly dependent (we cannot make any of them all $0's$.

So, Rank will be $\textbf{3}$.

**(A) is correct option!**

Everything is fine except “ None of the rows are linearly independent “. It should be

”none of the rows are linearly dependent “ so the rank is three because if rank is equal to :- number of linearly independent rows = number of linearly independent columns.

@Lakshman Patel RJIT Please edit this answer.

1

3 votes

Answer D. 4

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 1 | 4 | 8 | 7 |

2 | 0 | 0 | 3 | 0 |

3 | 4 | 2 | 3 | 1 |

4 | 3 | 12 | 24 | 2 |

Multiply the 1st row by 4. R1->R1×4

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 4 | 16 | 32 | 28 |

2 | 0 | 0 | 3 | 0 |

3 | 4 | 2 | 3 | 1 |

4 | 3 | 12 | 24 | 2 |

Subtract the 1st row from the 3rd row and restore it R3->R3-R1

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 1 | 4 | 8 | 7 |

2 | 0 | 0 | 3 | 0 |

3 | 0 | -14 | -29 | -27 |

4 | 3 | 12 | 24 | 2 |

Multiply the 1st row by 3. R1-->R1×3

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 3 | 12 | 24 | 21 |

2 | 0 | 0 | 3 | 0 |

3 | 0 | -14 | -29 | -27 |

4 | 3 | 12 | 24 | 2 |

Subtract the 1st row from the 4th row and restore it. R4-->R4-R1

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 1 | 4 | 8 | 7 |

2 | 0 | 0 | 3 | 0 |

3 | 0 | -14 | -29 | -27 |

4 | 0 | 0 | 0 | -19 |

Swap the 2nd and the 3rd rows R2<->R3

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 1 | 4 | 8 | 7 |

2 | 0 | -14 | -29 | -27 |

3 | 0 | 0 | 3 | 0 |

4 | 0 | 0 | 0 | -19 |

Calculate the number of linearly independent rows

A1 | A2 | A3 | A4 | |
---|---|---|---|---|

1 | 1 | 4 | 8 | 7 |

2 | 0 | -14 | -29 | -27 |

3 | 0 | 0 | 3 | 0 |

4 | 0 | 0 | 0 |
-19 |