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Consider a system of equations (λ – a)x + 2y +3z = 0, x +2(λ – b) y + 3z = 0, x + 2y + 3(λ – c) z = 0, which has a non-trivial solution.  

Product of all values of λ for above system is

(1) abc + a + b + c + 2
(2) abc + a + b + c – 2
(3) abc – a – b – c – 2
(4) abc – a – b – c + 2

 Ans given is Option C.

Can anyone explain the complete solution?

in Linear Algebra 261 views

3 Answers

2 votes
$A=\begin{bmatrix} \lambda- a &2 &3 \\ 1& 2\lambda-2b &3 \\ 1&2 & 3\lambda-3c \end{bmatrix}$

$=\begin{bmatrix} -a+\lambda &2 &3 \\ 1&-2b+2\lambda &3 \\ 1&2 & -3c+3\lambda \end{bmatrix}$

 

We find the values of $\lambda$ which satisfy the characteristic equation of the matrix $A$, namely those values of $\lambda$ for which

$|A-\lambda I|=0$

 

$\begin{vmatrix} -a+\lambda &2 &3 \\ 1&-2b+2\lambda &3 \\ 1&2 & -3c+3\lambda \end{vmatrix}=0$

$\begin{vmatrix} -a+\lambda &2+0 &3+0 \\ 1+0&-2b+2\lambda &3+0 \\ 1+0&2+0 & -3c+3\lambda \end{vmatrix}=0$

$\begin{vmatrix} -a &2 &3 \\ 1&-2b &3 \\ 1&2 & -3c \end{vmatrix}+\begin{vmatrix} \lambda &0 &0 \\ 0& 2\lambda &0 \\ 0&0 & 3\lambda \end{vmatrix}=0$

$\begin{vmatrix} -a &2 &3 \\ 1&-2b &3 \\ 1&2 & -3c \end{vmatrix}+6\lambda ^{3}\begin{vmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0&0 & 1 \end{vmatrix}=0$

$-6abc+6a+6b+6c+12+6\lambda ^{3}=0$

$6\lambda ^{3}=6abc-6a-6b-6c-12$

$\lambda ^{3}=abc-a-b-c-2$

 

Hence,Option$(3)abc-a-b-c-2$.
0

Thank you @LeenSharma Veteran.

You made it look so simple.

0 votes

M=$\begin{pmatrix} -a+\lambda &2 &3 \\ 1&-2b+2\lambda &3 \\ 1&2 & -3c+3\lambda \end{pmatrix}$=0

can be written as:-

M=$\begin{pmatrix} -a &2 &3 \\ 1&-2b &3 \\ 1&2 & -3c \end{pmatrix}$ / -6   -  $\lambda$/-6 $\begin{pmatrix} -1 &0 &0 \\ 0&-2 &0 \\ 0&0 & -3 \end{pmatrix}$=0

|M'-$\lambda$I|=0

:represent  $\lambda$ is eigen values of given matrix.....and product of all the values of is equal to the determinent of matrix M' /-6

M'=$\begin{pmatrix} -a &2 &3 \\ 1&-2b &3 \\ 1&2 & -3c \end{pmatrix}$   /-6

Det(M')devided by -6=option C


edited by
0

How did you split the matrix into two matrices?

@hs_yadav can you explain it further, pl.

0

stanchion

i had made a mistake .....now it's ok.....

0 votes
If system has a non-trivial solution then determinant of matrix should be Zero.

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