91Ó°ÊÓ

If A is an $\mathit{n}\mathbf{×}\mathit{m}$matrix with row vectors $\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{1}}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{n}}}\mathbf{}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{x}}$is a vector in $R^m$ then,

$\mathit{A}\stackrel{\mathbf{→}}{\mathbf{X}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{-}\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{1}}}\mathbf{-}\\ \mathbf{â‹\circledR }\\ \mathbf{-}\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{n}}}\mathbf{-}\end{array}\mathbf{\right]}\stackrel{\mathbf{→}}{\mathbf{X}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{-}\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{1}}}\mathbf{.}\stackrel{\mathbf{→}}{\mathbf{X}}\mathbf{-}\\ \mathbf{â‹\circledR }\\ \mathbf{-}\stackrel{\mathbf{→}}{{\mathbf{Ó\neg }}_{\mathbf{n}}}\mathbf{.}\stackrel{\mathbf{→}}{\mathbf{X}}\mathbf{-}\end{array}\mathbf{\right]}$

The given matrices are, $\stackrel{→}{x}=\left[\begin{array}{c}5\\ 3\\ -9\end{array}\right]$ and $\stackrel{→}{y}=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]$.

The matrix A with all nonzero entries is, role="math" localid="1664196291881" $\left[\begin{array}{ccc}x& a& b\\ d& y& c\\ e& f& z\end{array}\right]$.

The system $A\stackrel{→}{x}=\stackrel{→}{y}$is,

role="math" localid="1664196402249" $$\begin{gathered} \left[\begin{array}{ccc}x& a& b\\ d& y& c\\ e& f& z\end{array}\right]\left[\begin{array}{c}5\\ 3\\ -9\end{array}\right]=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right] \\ \left[\begin{array}{c}5x+3a-9b\\ 5d+3y-9c\\ 5e+3f-9z\end{array}\right]=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right] \end{gathered}$$

Represent the obtained matrix in form of equations.

$$\begin{gathered} 5x+3a-9b=2       ......\left(1\right) \\ 5d+3y-9c=0       ......\left(2\right) \\ 5e+3f-9z=1        ......\left(3\right) \end{gathered}$$

Consider the equation (1)

$5x+3a-9b=2 $

The values of ‘a’ and ‘b’ should be chosen such that, it satisfies the equation.

Let $a=2,b=1$, then,

$$\begin{gathered} 5x+3\left(2\right)-9\left(1\right)=2  \\ 5x+6-9=2 \\ 5x=5 \\ ∴x=1 \end{gathered}$$

Consider the equation (2)

$5d+3y-9c=0$

The values of ‘c’ and ‘d’ should be chosen such that, it satisfies the equation.

Let $c=4,d=6$, then,

$$\begin{gathered} 5\left(6\right)+3y-9\left(4\right)=0 \\ 30+3y-36=0 \\ 3y=6 \\ ∴y=2 \end{gathered}$$

Consider the equation (3)

$5e+3f-9z=1$

The values of ‘e’ and ‘f’ should be chosen such that, it satisfies the equation.

Let $e=\frac{1}{5},f=2$, then,

$$\begin{gathered} 5\left(\frac{1}{5}\right)+3\left(2\right)-9z=1 \\ 1+6-9z=1 \\ 9z=6 \\ ∴z=\frac{2}{3} \end{gathered}$$

The matrix A with all nonzero entries is,$\left[\begin{array}{ccc}x& a& b\\ d& y& c\\ e& f& z\end{array}\right]$ .

Substitute the values of x, y and z.

$\left[\begin{array}{ccc}x& a& b\\ d& y& c\\ e& f& z\end{array}\right]$=$\left[\begin{array}{ccc}1& 2& 1\\ 6& 2& 4\\ \frac{1}{5}& 2& \frac{2}{3}\end{array}\right]$

Hence, $\left[\begin{array}{ccc}1& 2& 1\\ 6& 2& 4\\ \frac{1}{5}& 2& \frac{2}{3}\end{array}\right]$is a matrix A with all nonzero entries such that $A\stackrel{→}{x}=\stackrel{→}{y}$.