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  • For which of the following ordered pairs , the system of linear equations <img alt="x+2y+3z=1" src="https://learn.careers360.com/latex-image/?x&plu

For which of the following ordered pairs (\mu ,\delta), the system of linear equations x+2y+3z=1 3x+4y+5z=\mu 4x+4y+4z=\varrho is inconsistent ?
Option: 1 (4,6)
Option: 2 (3,4)
Option: 3 (1,0)
Option: 4 (4,3)
 

Answers (1)

best_answer

 

 

Solution of System of Linear Equations Using Matrix Method -

let us consider n linear equations in n unknowns, given as below

 

\\\mathrm{a_{11}x_1+a_{12}x_2+...+a_{1n}x_n = b_1} \\\mathrm{a_{21}x_1+a_{22}x_2+...+a_{2n}x_n = b_2} \\\mathrm{...\;\;\;...\;\;\;...\;\;\;...\;\;\;\;\;\;\;\;...\;\;\;\;\;\;...} \\\mathrm{...\;\;\;...\;\;\;...\;\;\;...\;\;\;\;\;\;\;\;...\;\;\;\;\;\;...} \\\mathrm{a_{n1}x_1+a_{n2}x_2+...+a_{nn}x_n = b_n} \\\mathrm{here \; x_{1}, x_2,...x_n \; are \;unknown\; variables} \\\mathrm{if \; b_1=b_2 =...=b_n=0\; then\; the\; system \; of \; equation \; is} \\\mathrm{known\; as \; homogenous\; system \; of \; equation\; and \;if } \\\mathrm{any \; of \;b_1,b_2,...b_n \; is\; not \; known\; then \; it \; is \;called\; } \\\mathrm{non\; homogenous\; system \; of \; equation}

The above system of equations can be written in matrix form as 

 

\\\mathrm{\begin{bmatrix} a_{11} & a_{12} & ... & ... & a_{1n}\\ a_{21} & a_{22} & ... & ... & a_{2n}\\ ... & ... & ... & ... & ...\\ ... & ... & ... & ... & ...\\ a_{n1} & a_{n2} & ... & ... & a_{nn} \end{bmatrix}\begin{bmatrix} x_1\\ x_2\\ ...\\ ...\\ x_n \end{bmatrix} = \begin{bmatrix} b_1\\ b_2\\ ...\\ ...\\ b_n \end{bmatrix}} \\\\\mathrm{\Rightarrow AX = B, \; where} \\\\\mathrm{A = \begin{bmatrix} a_{11} & a_{12} & ... & ... & a_{1n}\\ a_{21} & a_{22} & ... & ... & a_{2n}\\ ... & ... & ... & ... & ...\\ ... & ... & ... & ... & ...\\ a_{n1} & a_{n2} & ... & ... & a_{nn} \end{bmatrix} , X=\begin{bmatrix} x_1\\ x_2\\ ...\\ ...\\ x_n \end{bmatrix}, B=\begin{bmatrix} b_1\\ b_2\\ ...\\ ...\\ b_n \end{bmatrix}}

 

Premultiplying equation AX=B by A-1, we get

     A-1(AX) = A-1B ⇒ (A-1A)X = A-1B

                     ⇒ IX = A-1B

                     ⇒  X = A-1B 

                      ⇒   \mathrm{X=\frac{adj A}{\left | A \right |}B}

 

Types of equation :

  1. System of equations is non-homogenous:

    1. If |A| ≠ 0, then the system of equations is consistent and has a unique solution X = A-1B

    2. If |A| = 0 and   (adj A)·B ≠ 0, then the system of equations is inconsistent and has no solution.

    3. If |A|  = 0 and   (adj A)·B = 0, then the system of equations is consistent and has infinite number of solutions.

  2. System of equations is homogenous:

    1. If |A| ≠ 0, then the system of equations has only trivial solution and it has one solution.

    2. If |A| = 0 then the system of equations has non-trivial solution and it has an infinite number of solution.

    3. If number of equation < number of unknown then it has non-trivial solution.

-

 

 

\begin{aligned} D &=\left|\begin{array}{lll}{3} & {4} & {5} \\ {1} & {2} & {3} \\ {4} & {4} & {4}\end{array}\right| \\ &=2\left|\begin{array}{lll}{3} & {4} & {5} \\ {1} & {2} & {3} \\ {2} & {2} & {2}\end{array}\right| \end{aligned}

\\R_1\rightarrow R_1-(R_2+R_3)\\D=\left|\begin{array}{lll}{3} & {4} & {5} \\ {1} & {2} & {3} \\ {2} & {2} & {2}\end{array}\right|=0

Hence, it has infinitely many solutions

\mathrm{P}_{3} \equiv \alpha \mathrm{P}_{1}+\beta \mathrm{P}_{2}

3 \alpha+\beta=4\;\; \&\;\; 4 \alpha+2 \beta=4 \Rightarrow \alpha=2 \;\; \&\;\; \beta=-2

2 \mu-2=\delta

For non infinite solution 2 \mu-2\neq\delta

Correct Option (4)

Posted by

Kuldeep Maurya

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