Question

Determine whether the statements that follow are true or false, and justify your answer.

19. There exits a matrix A such that$\mathbf{A}\left[\begin{array}{c}-1\\ 2\end{array}\right]\mathbf{=}\left[\begin{array}{c}3\\ 5\\ 7\end{array}\right]$

Short answer

It is true that there exists a matrix A such that$\mathrm{A}\left[\begin{array}{c}-1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 5\\ 7\end{array}\right]$

Step-by-step solution

Step 1:Assuming the matrix 

Since, the order of matrix A should be $3\times 2$

Let the matrix A be

$\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]$

Justification of answer  

Now put the value of Matrix A in the given equation we get.

$\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]\left[\begin{array}{c}-1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 5\\ 7\end{array}\right]$

Now after solving the above equation we get.

$$\begin{gathered} -a+2b=3 \\ -c+2d=5 \\ -e+2f=7 \\ \Rightarrow b=\frac{3+a}{2},d=\frac{5+c}{2},f=\frac{7+e}{2} \end{gathered}$$

The A matrix will become

$\left[\begin{array}{cc}a& \frac{3+a}{2}\\ c& \frac{5+c}{2}\\ e& \frac{7+e}{2}\end{array}\right]$

Hence, it is true that there exists a matrix A such that$\mathrm{A}\left[\begin{array}{c}-1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 5\\ 7\end{array}\right]$