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) (...

Answer Details

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\(^{-1}\) = \(\frac{Adj (R)}{|R|}\) = \(\frac{1}{11}\)\(\begin{pmatrix} 5  & 2 \\ -3 & 1 \end{pmatrix}\)

= \(\begin{pmatrix} \frac{5}{11}  & \frac{2}{11} \\ \frac{-3}{11} & \frac{1}{11} \end{pmatrix}\)