(a) Write down the matrix A of the linear transformation \(A(x, y) \to (2x -y, -5x + 3y)\).
(b) If \(B = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\), find :
(i) \(A^{2} - B^{2}\) ; (ii) matrix \(C = B^{2} A\) ; (iii) the point \(M(x, y)\) whose image under the linear transformation \(C\) is \(M' (10, 18)\).
(c) What is the relationship between matrix A and matrix C?
Show Answer Show Explanation Explanation (a) \(A(x, y) \to (2x - y , -5x + 3y)\) Matrix \(A = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}\) (b) \(A = \begin{pmatrix} 2 & -1 \\ -5 & 2 \end{pmatrix} ; B = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\) (i) \(A^{2} - B^{2} = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} \begin{pmatrix} 2 & -1 \\ -5 & 2 \end{pmatrix} - \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix} \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\) = \(\begin{pmatrix} 9 & -5 \\ -25 & 14 \end{pamtrix} - \begin{pmatrix} 14 & 5 \\ 25 & 9 \end{pmatrix}\) = \(\begin{pmatrix} -5 & -10 \\ -50 & 5 \end{pmatrix}\) = \(-5 \begin{pmatrix} 1 & 2 \\ 10 & -1 \end{pmatrix}\) (ii) \(C = B^{2} A\) = \(\begin{pmatrix} 14 & 5 \\ 25 & 9 \end{pmatrix} \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}\) = \(\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}\) (iii) \(\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 10 \\ 18 \end{pmatrix}\) \(3x + y = 10 ... (1)\) \(5x + 2y = 18 ... (2)\) Multiply (1) by 2 : \(6x + 2y = 20 ... (3)\) \((3) - (2) : (6x + 2y) - (5x + 2y) = (20 - 18)\) \(x = 2\) Put x = 2 in (1) : \(3x + y = 10 \implies 3(2) + y = 10\) \(6 + y = 10 \implies y = 10 - 6 = 4\) \(M(2, 4)\). (c) C is the inverse of matrix A and A is the inverse of matrix C.