Three linear transformations, P, Q, and R in the oxy plane are defined by
P: (x, y) → (-4x – y, 2x)
Q: (x, y) → (y, 6x – 9y)
R: (x, y) → (x – 2y, 3x + 5y)
(a) write down the matrices of P, Q, and R
(b) Find:
(i) 2P – 3R + Q;
(ii) QR;
(iii) the inverse of the matrix R.
Explanation
Given:
P: (x, y) → (-4x - y, 2x)
Q: (x, y) → (y, 6x - 9y)
R: (x, y) → (x - 2y, 3x + 5y)
(a) the matrix of P, Q, and R
P = \(\begin{pmatrix} - 4 & -1 \\ 2 & 0 \end{pmatrix}\), Q = \(\begin{pmatrix} 0 & 1 \\ 6 & -9 \end{pmatrix}\), R = \(\begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix}\),
(b) (i) 2P - 3R + Q
= 2\(\begin{pmatrix} - 4 & -1 \\ 2 & 0 \end{pmatrix}\) - 3\(\begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix}\) + \(\begin{pmatrix} 0 & 1 \\ 6 & -9 \end{pmatrix}\)
= \(\begin{pmatrix} -8 & -2 \\ 4 & 0 \end{pmatrix}\) - \(\begin{pmatrix} 3 & -6 \\ 9 & 15 \end{pmatrix}\) + \(\begin{pmatrix} 0 & 1 \\ 6 & -9 \end{pmatrix}\)
= \(\begin{pmatrix} -11 & 5 \\ 1 & -24 \end{pmatrix}\)
(ii) QR = \(\begin{pmatrix} 0 & 1 \\ 6 & -9 \end{pmatrix}\)\(\begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix}\)
= \(\begin{pmatrix} 0 + 3 & 0+ 5 \\ 6 - 27 & -12 - 45 \end{pmatrix}\)
= \(\begin{pmatrix} 3 & 5 \\ -21 & -57 \end{pmatrix}\)
(iii) the inverse of the matrix R. \(\begin{pmatrix} 1 & -2 \\ 3 & 5 \end{pmatrix}\),
|R| = (1 x 5) - ( -2 x 3) = 5 + 6 = 11
R\(^{-1}\) = \(\frac{Adj (R)}{|R|}\) = \(\frac{1}{11}\)\(\begin{pmatrix} 5 & 2 \\ -3 & 1 \end{pmatrix}\)
= \(\begin{pmatrix} \frac{5}{11} & \frac{2}{11} \\ \frac{-3}{11} & \frac{1}{11} \end{pmatrix}\)