Question

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

16: There exists a 2x2 matrix such that

$\mathit{A}\left[\begin{array}{c}1\\ 1\end{array}\right]\mathbf{=}\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{A}\left[\begin{array}{c}2\\ 2\end{array}\right]\mathbf{=}\left[\begin{array}{c}2\\ 1\end{array}\right]$.

Short answer

Answer:

False, there doesn’t exist a 2x2 matrix A such that $A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]andA\left[\begin{array}{c}2\\ 2\end{array}\right]=\left[\begin{array}{c}2\\ 1\end{array}\right]$.

Step-by-step solution

Taking the matrix 

Suppose the matrix A is$A\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.

Now according to the given equations

$A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]andA\left[\begin{array}{c}2\\ 2\end{array}\right]=\left[\begin{array}{c}2\\ 1\end{array}\right]$ .

Justification of answer

Put the value of A matrix in the above equation we get.

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]$

After solving the equation we get

$\left[\begin{array}{c}a+b\\ c+d\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]$

Now on equating the value of matrix we get

a + b = 1

c + d = 1

We also have another equation.

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}2\\ 2\end{array}\right]=\left[\begin{array}{c}2\\ 1\end{array}\right]$

After solving the equation we get

$\left[\begin{array}{c}21+2b\\ 2c+2d\end{array}\right]=\left[\begin{array}{c}2\\ 1\end{array}\right]$

Now on equating the value of matrix we get

a + b = 1

2c + 2d = 1

On solving the equations, we get the value of variables

Since, the above equations can’t be solved.

Hence, there doesn’t exist a 2x2 matrix A such that $A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]andA\left[\begin{array}{c}2\\ 2\end{array}\right]=\left[\begin{array}{c}2\\ 1\end{array}\right]$.