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Determine the value of A & B for which system $$
\begin{pmatrix}
3 & -2 & 1 \\
5 & -8 & 9 \\
2 & 1 & A \\
\end{pmatrix} \begin{pmatrix}
x \\
y \\
z \\
\end{pmatrix}=\begin{pmatrix}
B \\
3 \\
-1 \\
\end{pmatrix}
$$
1) has a unique solution. |
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Note that an equation of the form $\mathbf{A}\vec{x}=\vec{b}$ has a unique solution iff $\det\mathbf{A}\not=0$, therefore, we can find the value of $\mathbf{A}$ for which there is a unique solution: $$\det\mathbf{A}=\begin{vmatrix}3 & -2 & 1 \\ 5 & -8 & 9 \\ 2 & 1 & A \end{vmatrix}\not= 0\implies-14A-42\not=0 \therefore A\not=-3$$ If $A\not=3$ then $B\in\mathbb{R}$ is a sufficient condition to ensure a unique solution. If there are an infinite number of solutions, then $A=-3$ and $3x-2y+z=B$ is some linear combination of $5x-8y+9z=3$ and $2x+y-3z=-1$. Solving the following system: $$\begin{pmatrix}3 \\ -2 \\ 1\end{pmatrix}=\lambda\begin{pmatrix}5 \\ -8 \\ 9\end{pmatrix}+\mu\begin{pmatrix}2 \\ 1 \\ -3\end{pmatrix}$$ Gives the solutions $\lambda=\frac{1}{3}$ and $\mu=\frac{2}{3}$, therefore $B=1-\frac{2}{3}=\frac{1}{3}$ for infinite solutions. If there are no solutions then the three equations are inconsistent and therefore $B\not=\frac{1}{3}$. |
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compare the ranks of the matrices $\begin{pmatrix} 3 & -2 & 1 \\ 5 & -8 & 9 \\ 2 & 1 &A \end{pmatrix}$ and $\begin{pmatrix} 3 & -2 & 1 & B \\ 5 & -8 & 9 &3 \\ 2 & 1 &A&-1 \end{pmatrix}$. If the ranks are equal, there is a solution. If they are not, there is none. There is a unique solution iff the ranks are full, i.e. $3$. The interisting values should be $A=-3$ and $B=\frac 13$. |
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For 1, it is proved that the non-homogeneous system of $n$ equations with $n$ variables has unique solution if and only if the matrix including all coefficient of the system has a non zero determinant. So evaluate the determine of $$ S:=\begin{pmatrix} 3 & -2 & 1 \\ 5 & -8 & 9 \\ 2 & 1 & A \\ \end{pmatrix}$$ and put it equal to zero. The resulted values should not be given to $A$. Here we have $\det(S)=-14A-42$ and so for all $A\neq\frac{42}{-14}$ the system has unique solution. |
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