# Matrices Long Questions (Question 4)

Question 4:
It is given that  $Q\left(\begin{array}{cc}3& -2\\ 5& -4\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ , where Q is a 2 × 2 matrix.
(a) Find the matrix Q.
(b) Write the following simultaneous linear equations as matrix equation:
3x – 2y = 7
5x – 4y = 9
Hence, using matrix method, calculate the value of x and y.

Solution:
(a)
$\begin{array}{l}Q=\frac{1}{3\left(-4\right)-\left(5\right)\left(-2\right)}\left(\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right)\\ =-\frac{1}{2}\left(\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right)\\ =\left(\begin{array}{cc}2& -1\\ \frac{5}{2}& -\frac{3}{2}\end{array}\right)\end{array}$
(b)