# I want to know how to find all the values of k for which the equation Ax=b has a unique solution for every 3x1 matrix b?Consider the matrix: K -1 0 A = 2 K -1 2 1 2...

I want to know how to find all the values of k for which the equation A**x=b **has a unique solution for every 3x1 matrix **b**?

Consider the matrix:

K | -1 | 0 | |

A = | 2 | K | -1 |

2 | 1 | 2 |

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The requirement is that the system equations represented by Ax = b have unique solution and conditions if any on value of k .

This possible in case of a square matrix if all the 3 equations by Ax = b are independent or the rank of the given matrix is 3. This is possible only if the determinant of A is not equal to zero.

Therefore ||A|| nort zero.

Now let us evaluate ||A|| interms of 1st row:

Det A = k( k*2 - -1) - (-1) (2*2- -1*2) +0*(... )

Det A = k(2k+1) + 6 .

Det A = 2k^2 +k +(6) .

DetA = 2{k^2+k/2 } + 6

DetA = 2(k^2 +k/2 +1/4) - 2/4 +6

DetA = 2(k+1/2)^2 + 11/2 , first term 2(+1/2)^2 is 2 times the square expression is positive for all real k. The 2nd term 11/2 is also a positive. So therminant A is positive and is never zero. Therefore the solution of the system of equations Ax = b has unique solution for all real values of k.