Question

There exists a$\mathbf{2}\mathbf{\times }\mathbf{2}$matrix such that$\mathit{A}\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{=}\left[\begin{array}{c}3\\ 4\end{array}\right]$

Short answer

True, there exist a$2\times 2$ matrix A such thatrole="math" localid="1659642050851" $A\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Step-by-step solution

Step 1: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 equation,

$A\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Justification of answer

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

$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$

After solving the equation we get

$\left[\begin{array}{c}a+2b\\ c+2d\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Now on equating the value of matrix we get

$$\begin{gathered} a+2b=3 \\ c+2d=4 \end{gathered}$$

On solving the equations we get the value of variables

$$\begin{gathered} a=3-2b \\ c=4-2d \end{gathered}$$

Where, are free variables which can have any value. For instance let the value of$b,d=1$

It implies$a=1,c=2$

Thus, we get the matrix

$A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

Hence, it is truethat there exist a $2\times 2$matrix A such that$A\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}3\\ 4\end{array}\right]$