Animated Solution for Mathematics - Matrices and Determinants: The system of equations $kx+y+z=1,x+ky+z=k$ and $x+y+zk=k^2$ has no solution if $k$ is equal to:

We are investigating the system of linear equations defined by:

We seek the value of $k$ for which this system is inconsistent, meaning there is no set of $(x,y,z)$ that satisfies all three equations simultaneously.

The Determinant $\Delta$

The existence of a unique solution is governed by the determinant of the coefficient matrix, denoted as $\Delta$. If $\Delta eq 0$, the system has a unique solution. If $\Delta =0$, the system is either inconsistent or has infinitely many solutions.

We calculate the determinant as follows:

Expanding along the first row:

Simplifying the expression leads to:

Factoring this cubic polynomial yields:

Setting $\Delta =0$, we identify the critical values: $k=1$ and $k=-2$.

We must verify if these values lead to inconsistency or infinite solutions. Let us test $k=1$.

Substituting $k=1$ into the original system, all three equations collapse into the identical form:

Since all three planes are coincident, the system has infinitely many solutions. Therefore, $k=1$ is not the value we are looking for.

Testing $k=-2$

We now test $k=-2$. According to Cramer's Rule, if $\Delta =0$ and at least one of the auxiliary determinants (such as ${\Delta }_1$) is non-zero, the system is inconsistent.

We define ${\Delta }_1$ by replacing the first column of the coefficient matrix with the constants $(1,k,k^2)$:

Substituting $k=-2$ into this determinant:

Calculating the value:

Since $\Delta =0$ and $\Delta_1 = 9 eq 0$, the system is confirmed to be inconsistent. The final value that renders the system inconsistent is $k=-2$.