If $x\left[ \begin{array}{l} - 3\\4\end{array} \right] + y\left[ \begin{array}{l}4\\3\end{array} \right] = \left[ \begin{array}{l}10\\ - 5\end{array} \right]$. Then what are the values of $x$ and $y$?
A. $x = - 2, y = 1$
B. $x = - 9, y = 10$
C. $x = 22, y = 1$
D. $x = 2, y = - 1$

Hint: Here simplify the left-hand side by using the scalar multiplication property of a matrix. Then use the additional property of matrices. After that, use the property of equality of matrix and equate the corresponding elements. In the end, solve the equations with variables $x$ and $y$ to reach the required answer.

Formula Used:
The scalar multiplication of a matrix is a product of a matrix and a real number.
Equality property: Two matrices are equal if and only if the matrices have the same order and the corresponding elements are identical.

Complete step by step solution:
 The given equation of matrices is $x\left[ \begin{array}{l} - 3\\4\end{array} \right] + y\left[ \begin{array}{l}4\\3\end{array} \right] = \left[ \begin{array}{l}10\\ - 5\end{array} \right]$.
Let’s simplify the above equation.
Apply the scalar multiplication property of a matrix.
$x\left[ \begin{array}{l} - 3\\4\end{array} \right] + y\left[ \begin{array}{l}4\\3\end{array} \right] = \left[ \begin{array}{l}10\\ - 5\end{array} \right]$
$ \Rightarrow $$\left[ \begin{array}{l} - 3x\\4x\end{array} \right] + \left[ \begin{array}{l}4y\\3y\end{array} \right] = \left[ \begin{array}{l}10\\ - 5\end{array} \right]$
Now apply the additional property of matrices.
$\left[ \begin{array}{l} - 3x + 4y\\4x + 3y\end{array} \right] = \left[ \begin{array}{l}10\\ - 5\end{array} \right]$

Apply equality property of matrices.
We get,
$ - 3x + 4y = 10$ $.....\left( 1 \right)$
$4x + 3y = - 5$ $.....\left( 2 \right)$

Multiply equation $\left( 1 \right)$ by 4 and multiply equation $\left( 2 \right)$ by 3.
$ - 12x + 16y = 40$
$12x + 9y = - 15$
Now add the above equations.
$25y = 25$
Divide both sides by $25$.
$y = 1$

Substitute $y = 1$ in the equation $\left( 1 \right)$.
$ - 3x + 4\left( 1 \right) = 10$
$ \Rightarrow $$ - 3x + 4 = 10$
$ \Rightarrow $$ - 3x = 6$
Divide both sides by $ - 3$.
$x = - 2$
Thus, the values are $x = - 2$, and $y = 1$.

Option ‘A’ is correct

Note: Many students make a mistake to multiply $x$ and $y$ with $\left[ \begin{array}{l} - 3\\4\end{array} \right]$ and $\left[ \begin{array}{l}4\\3\end{array} \right]$ respectively. They multiply it with first row which is incorrect way. In scalar multiplication, each element of the matrix is multiplied by the given scalar. So $x$ and $y$will be multiplied with each elements of $\left[ \begin{array}{l} - 3\\4\end{array} \right]$ and $\left[ \begin{array}{l}4\\3\end{array} \right]$ respectively.