(a) Write down the matrix A of the linear transformation \(A(x, y) \to (2x -y, -5x + 3y)\). (b) If \(B = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\), fin...

Answer Details

(a) The transformation \(A(x,y)\to(2x - y,\, -5x + 3y)\) has matrix

\[A = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}.\]

(b) First compute the squares.

\[A^2 = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 9 & -5 \\ -25 & 14 \end{pmatrix},\qquad B^2 = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}^2 = \begin{pmatrix} 14 & 5 \\ 25 & 9 \end{pmatrix}.\]

(i) \(\displaystyle A^2 - B^2 = \begin{pmatrix} 9-14 & -5-5 \\ -25-25 & 14-9 \end{pmatrix} = \begin{pmatrix} -5 & -10 \\ -50 & 5 \end{pmatrix}.\)

(ii) \(\displaystyle C = B^2 A = \begin{pmatrix} 14 & 5 \\ 25 & 9 \end{pmatrix}\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}.\)

(iii) If \(C\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}10\\18\end{pmatrix}\), then \(3x + y = 10\) and \(5x + 2y = 18\).

From the first, \(y = 10 - 3x\); substituting: \(5x + 2(10-3x) = 18 \Rightarrow -x + 20 = 18 \Rightarrow x = 2\), then \(y = 4\). So \(M(2, 4)\).

(c) Note \(\det A = (2)(3)-(-1)(-5) = 1\), and \(AC = \begin{pmatrix}1&0\\0&1\end{pmatrix} = I\). Hence \(C\) is the inverse of \(A\), that is \(C = A^{-1}\).