Chapter 11, Problem 24RE

To determine

To find: The value of $x$ , $y$ and $z$ .

Answer to Problem 24RE

The determine of matrix $A$ is zero $adj\left(A\right)\cdot \left(B\right)=0$

The system is inconsistent.

Explanation of Solution

Given information:

  $\begin{array}{l}x-y+z=0\\ x-y-5z-6=0\\ 2x-2y+z-1=0\end{array}$

Calculation:

As

  $\begin{array}{l}x-y+z=0\\ x-y-5z=6\\ 2x-2y+z=1\end{array}$

In mathematics

  $\begin{array}{l}\left[\begin{array}{l}1-11\\ 1-1-5\\ 2-21\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}0\\ 6\\ 1\end{array}\right]\\ \Rightarrow AX=B\\ \Rightarrow X=A^{-1}B---\left(i\right)\end{array}$

Now,

  $\begin{array}{l}{A}^{-1}=\frac{adj\left(A\right)}{\left|A\right|}\left[where,\left|A\right|\ne 0\right]\\ \Rightarrow \left|A\right|=1\left(-1-10\right)+\left(1+10\right)+1\left(-2+2\right)\\ \Rightarrow \left|A\right|=-11+11\\ \Rightarrow \left|A\right|=0\end{array}$

Now,

  $\begin{array}{l}adj\left(A\right)={\left(cofactor\text{ of }A\right)}^T\\ \Rightarrow cofactor\text{ of }A=\left[\begin{array}{l}+\left(-1-10\right)-\left(1+10\right)-\left(-2+2\right)\\ -\left(-1+2\right)+\left(1-2\right)-\left(-2+2\right)\\ +\left(5+1\right)-\left(-5-1\right)+\left(-1+1\right)\end{array}\right]\\ \Rightarrow cofactor\text{ of }A=\left[\begin{array}{l}-11-110\\ -1-10\\ 660\end{array}\right]\\ \end{array}$

  $\begin{array}{l}adj\left(A\right)={\left(cofactor\text{ of }\left(A\right)\right)}^T\\ =\left[\begin{array}{l}-11-16\\ -11-16\\ 000\end{array}\right]\end{array}$

  $\begin{array}{l}adj\left(A\right)\cdot \left(B\right)=\left[\begin{array}{l}-11-16\\ -11-16\\ 000\end{array}\right]\left[\begin{array}{l}0\\ 6\\ 1\end{array}\right]\\ \Rightarrow adj\left(A\right)\cdot \left(B\right)=\left[\begin{array}{l}-11\times 0+\left(-1\right)\text{6+}6\times 1\\ -11\times 0+\left(-1\right)6+6\times 1\\ 0\times 0+0\times 6+0\times 1\end{array}\right]\\ \Rightarrow adj\left(A\right)\cdot \left(B\right)=\left[\begin{array}{l}0\\ 0\\ 0\end{array}\right]\\ \Rightarrow adj\left(A\right)\cdot \left(B\right)=0\end{array}$

Hence, the determine of matrix $A$ is zero $adj\left(A\right)\cdot \left(B\right)=0$

Hence, the system is inconsistent.