18. $\mathrm { A } = \left[ \begin{array} { c c c } \mathrm { a } & 0 & 1 \\ 1 & \mathrm { c } & \mathrm { b } \\ 1 & \mathrm {~d} & \mathrm {~b} \end{array} \right] , \mathrm { B } = \left[ \begin{array} { c c c } \mathrm { a } & 1 & 1 \\ 0 & \mathrm {~d} & \mathrm { c } \\ \mathrm { f } & \mathrm { g } & \mathrm { h } \end{array} \right] , \mathrm { U } = \left[ \begin{array} { c } \mathrm { f } \\ \mathrm { g } \\ \mathrm { h } \end{array} \right] , \mathrm { V } = \left[ \begin{array} { c } \mathrm { a } ^ { 2 } \\ 0 \\ 0 \end{array} \right]$. If there is vector matrix X , such that $\mathrm { AX } = \mathrm { U }$ has infinitely many solutions, then prove that $\mathrm { BX } = \mathrm { V }$ cannot have a unique solution. If afd $\neq 0$ then prove that $\mathrm { BX } = \mathrm { V }$ has no solution.
Sol. $\mathrm { AX } = \mathrm { U }$ has infinite solutions $\Rightarrow | \mathrm { A } | = 0$ $\left| \begin{array} { c c c } \mathrm { a } & 0 & 1 \\ 1 & \mathrm { c } & \mathrm { b } \\ 1 & \mathrm {~d} & \mathrm {~b} \end{array} \right| = 0 \Rightarrow \mathrm { ab } = 1$ or $\mathrm { c } = \mathrm { d }$ and $\left| \mathrm { A } _ { 1 } \right| = \left| \begin{array} { c c c } \mathrm { a } & 0 & \mathrm { f } \\ 1 & \mathrm { c } & \mathrm { g } \\ 1 & \mathrm {~d} & \mathrm {~h} \end{array} \right| = 0 \Rightarrow \mathrm {~g} = \mathrm { h } ; \left| \mathrm { A } _ { 2 } \right| = \left| \begin{array} { c c c } \mathrm { a } & \mathrm { f } & 1 \\ 1 & \mathrm {~g} & \mathrm {~b} \\ 1 & \mathrm {~h} & \mathrm {~b} \end{array} \right| = 0 \Rightarrow \mathrm {~g} = \mathrm { h }$ $\left| \mathrm { A } _ { 3 } \right| = \left| \begin{array} { l l l } \mathrm { f } & 0 & 1 \\ \mathrm {~g} & \mathrm { c } & \mathrm { b } \\ \mathrm { h } & \mathrm { d } & \mathrm { b } \end{array} \right| = 0 \Rightarrow \mathrm {~g} = \mathrm { h } , \mathrm { c } = \mathrm { d } \Rightarrow \mathrm { c } = \mathrm { d }$ and $\mathrm { g } = \mathrm { h }$ $\mathrm { BX } = \mathrm { V }$ $| \mathrm { B } | = \left| \begin{array} { l l l } \mathrm { a } & 1 & 1 \\ 0 & \mathrm {~d} & \mathrm { c } \\ \mathrm { f } & \mathrm { g } & \mathrm { h } \end{array} \right| = 0 \quad$ (since $\mathrm { C } _ { 2 }$ and $\mathrm { C } _ { 3 }$ are equal) $\quad \Rightarrow \mathrm { BX } = \mathrm { V }$ has no unique solution. and $\left| \mathrm { B } _ { 1 } \right| = \left| \begin{array} { l l l } \mathrm { a } ^ { 2 } & 1 & 1 \\ 0 & \mathrm {~d} & \mathrm { c } \\ 0 & \mathrm {~g} & \mathrm {~h} \end{array} \right| = 0 ($ since $\mathrm { c } = \mathrm { d } , \mathrm { g } = \mathrm { h } )$ $\left| \mathrm { B } _ { 2 } \right| = \left| \begin{array} { c c c } \mathrm { a } & \mathrm { a } ^ { 2 } & 1 \\ 0 & 0 & \mathrm { c } \\ \mathrm { f } & 0 & \mathrm {~h} \end{array} \right| = \mathrm { a } ^ { 2 } \mathrm { cf } = \mathrm { a } ^ { 2 } \mathrm { df } \quad ($ since $\mathrm { c } = \mathrm { d } )$
$$\left| \mathrm { B } _ { 3 } \right| = \left| \begin{array} { c c c } \mathrm { a } & 1 & \mathrm { a } ^ { 2 } \\ 0 & \mathrm {~d} & 0 \\ \mathrm { f } & \mathrm {~g} & 0 \end{array} \right| = \mathrm { a } ^ { 2 } \mathrm { df }$$
since if $\operatorname { adf } \neq 0$ then $\left| \mathrm { B } _ { 2 } \right| = \left| \mathrm { B } _ { 3 } \right| \neq 0$. Hence no solution exist.