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Math Word Problem

Answered question

2022-03-21

Answer & Explanation

Vasquez

Expert2022-03-29Added 669 answers

Let the homogeneous system $A X = 0$ , be given by

$A X = [\begin{matrix} a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16} \\ a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26} \\ a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} \\ a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46} \end{matrix}] \begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \end{matrix} = 0$

(a) Fewest number of free varables occur, when all the four rows of coefficient matrix are linearly independent. In this case two of the six variables have arbitraty assignment of values (say $x_{5}$ and $x_{6}$ ). The remaining variables can be expressed in terms of these two variables (here $x_{5}$ and $x_{6}$ ).

$∴$ Fewest number of free varables $= 2$

(b) Maximum number of free varables occur, when all the four rows of coefficient matrix are linearly dependent. Since, at least one row is non-zero, the other three rows must be dependent to this row. In this case five of the six variables, have arbitraty assignment of values (say $x_{2}, x_{3}, x_{4}, x_{5} and x_{6}$ ). The remaining variable(s) (here $x_{1}$ ) can be expressed in terms of these five free variables (here $x_{2}, x_{3}, x_{4}, x_{5} and x_{6}$ ).

$∴$ Maximum number of free varables =5

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