Question: 3] a) Investigate for what values of $\mu$ and $\pi$ the equations x+ y+ z = 6, x + 2y + 3z = 10, x + 2y + $\pi$z = $\mu$ has
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1) No solution

2) A unique solution

3) Infinite number of solutions

Subject:- Applied Mathematics

Marks:- 3

Mumbai Unversity>FE>Sem1>Applied Maths1

m1(81) • 98 views
 modified 7 months ago  • written 7 months ago by
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we have $\begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 2 & \pi \end{bmatrix} $$\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 6 \ 10 \ \mu \end{bmatrix} By R_2- R_1, R_3- R_2 \begin{bmatrix} 1 & 1 & 1 \ 1 & 2 & 3 \ 1 & 2 & \pi-3 \end{bmatrix}$$ \begin{bmatrix} x \ y \ z \end{bmatrix}$ = $\begin{bmatrix} 6 \ 10 \ \mu-10 \end{bmatrix}$ (i) The system has unique solution if the coefficient matrix is non-singular (or the rank A, r= the number of unknowns , n =3). This requires λ-3 not equal to 0, Hence λ is not equal to 3. Hence the system has unique solution. (ii) If λ=3 the coefficient matrix and the augmented matrix becomes $\begin{bmatrix} 1 & 1 & 1 \ 0 & 1 & 2 \ 0 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} 1 & 1 & 1 & 6 \ 0 & 1 & 2 & 4 \ 0 & 0 & 0 & \mu-10 \end{bmatrix}$

The rank of A = 2 the rank of [A,B] will be also 2 if μ = 10.

Thus if λ=33and μ = 10 thesystem is consistent. But the rank of A (= 2) is less than the number of unknowns (=3). Hence the equation will posses infinite solutions.

(iii) If λ=3 and μ≠10 , the rank of A=2, and the rank of [A,B] = 3. They are not equal and the equations will be inconsistent and will not posses any solution.