MTH501 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || LINEAR ALGEBRA || BY VuTech

MTH501 ASSIGNMENT NO. 1 FALL 2022

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MTH501 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || LINEAR ALGEBRA || BY VuTech

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ASSIGNMENT#1

Fall 2022     

     Total marks: 15

     Due Date: 29^th November, 2022

          

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Question 1:                                                                                                               Marks = 05

Find a parametric equation of the line through `\bar u `  and `\bar v` , where

SOLUTION:

Given that,

let suppose that

Now, we find the direction vector 

Here

`a = 6, b =  - 7` c = 11

Now parametric equations are

`x = x_0 + at`

`\Rightarrow x =  - 6 + 6t`

`y = y_0 + bt`

`\Rightarrow y = 3 - 7t`

`z = z_0 + ct`

`\Rightarrow z =  - 4 + 11t`

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Question 2:                                                                                                               Marks = 10

Determine the value(s) of h and k such that the given system 

`x + 5y - 4z = 3,`

`3x + 12y - 12z =  - 3,`

`2x + 7y + hz = k`

has 

1.         no solution,

2.         a unique solution,

3.         many solutions.

 

SOLUTION:

Given\,system

`x + 5y - 4z = 3`

`3x + 12y - 12z =  - 3`

`2x + 7y + hz = k`

Now, in the matrix form

let,

So,

`AX = B`

Now, in the form of Augmented matric

As

`C = [A:B]`

Now, their consistent matrices are

Now, discuss the given conditions:

i.          NO SOLUTION:

`If  h =  - 8 and k \ne  - 6`

then `the\[R(A) < R(B)`, there will be no solution.

For that R(A)=2 and R(B)=3.

ii.         UNIQUE SOLUTION:

`If  h \ne  - 8 and k \ne  - 6`

then the R(A) = R(B) = 3 = no. of unknown, there will be unique solutions.

For that R(A)=3 and R(B)=3.

iii.        MANY SOLUTION:

If `h =  - 8 and k =  - 6` then the R(A) = R(B) = 2 < no. of unknown, there will be many solutions.

Like that R(A)=2 and R(B)=2.

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