Question 1062448
.
<pre>
Calculate the determinant of the matrix A = {{{(matrix(3,3, k,1,1, 1,k,1, 1,1,k))}}}.

If you are unfamiliar with it or want to refresh your knowledge, see the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A> 

in this site.


The determinant is equal to {{{k^3 + 1 + 1 - k -k - k}}} = {{{k^3 - 3k +2}}}.

The general theory says:


     If the determinant is not zero, then the system has a unique solution.

     The degenerate cases are those when the determinant is zero:

     d(k) = {{{k^3 - 3k  + 2}}} = 0.


One root of this equation is k = 1.

Hence, the polynomial is multiple of (k-1), and the ratio {{{d(k)/(k-1)}}} is {{{x^2 + x -1}}}.

Then the two other roots of the determinant are the zeroes of this quadratic polynomial {{{k^2 + k -1}}}. 

You can easily find these zeroes using the quadratic formula:

{{{k[1]}}} = {{{(-1 + sqrt(5))/2}}}  and  {{{k[1]}}} = {{{(-1 - sqrt(5))/2}}}.


Next, if k = 1, your original system consists of three identical equations.
Hence, the system has infinitely many solutions at k = 1.

If k = {{{k[1]}}} or k = {{{k[2]}}}, then the system has infinitely many solutions, too.

   Indeed, if you add all three equations at these values of k, you will get the equation 

   (k+2)x + (k+2)y + (k+2)z = 3,  or

   x + y + z = {{{3/(k+2)}}}.   (*)

   Now distract      the equation (1) from (*), and you will find the solution for x.
   Distract then     the equation (2) from (*), and you will find the solution for y.
   Finally, distract the equation (3) from (*), and you will find the solution for z.

   Thus the system has at least one solution for k = {{{k[1]}}} and/or k = {{{k[2]}}}, having the determinant equal to zero at these values of k.
    It is enough for the system to have infinitely many solutions then.
</pre>

<U>Answer</U>. The system has infinitely many solutions at k = 1, k = {{{(-1 + sqrt(5))/2}}}  and  k = {{{(-1 - sqrt(5))/2}}}.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The system has a unique solution for all other values of k.



------------------------------
The useful references on this theme are these lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/HOW-TO-solve-system-of-linear-eqns-in-three-unknowns-using-det.lesson>HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-three-unknowns-using-determinant.lesson>Solving systems of linear equations in three unknowns using determinant (Cramer's rule)</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
&nbsp;&nbsp;&nbsp;&nbsp; "<U>3x3-Matrices, determinants, Cramer's rule for systems in three unknowns</U>"