Draw triangle PQR whose vertices are P(1,1), Q(-3,2) and R(0,3) on the grid provided. And draw the image P'Q'R', image of APQR under the transformation whose matrix is [[-1,0],[0,1]]. Find the co-ordinates of P'Q'R' and draw the image. Also find the matrix that maps P'Q'R' onto PQR.

Kwa P(1,1):

$$P' = \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} (2)(1) + (0)(1) \\ (-1)(1) + (1)(1) \end{pmatrix} = \begin{pmatrix} 2 + 0 \\ -1 + 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$$

Kwa Q(3,2):

$$Q' = \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} (2)(3) + (0)(2) \\ (-1)(3) + (1)(2) \end{pmatrix} = \begin{pmatrix} 6 + 0 \\ -3 + 2 \end{pmatrix} = \begin{pmatrix} 6 \\ -1 \end{pmatrix}$$

Kwa R(0,3):

$$R' = \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} (2)(0) + (0)(3) \\ (-1)(0) + (1)(3) \end{pmatrix} = \begin{pmatrix} 0 + 0 \\ 0 + 3 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}$$

Vipeo vya picha P'Q'R' ni:

• P'(2,0)
• Q'(6,-1)
• R'(0,3) Chora pointi hizi kwenye gridi na uziunganishe kuunda pembetatu P'Q'R'.

Kwa P'(2,0):

$$P'' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 \\ 0 \end{pmatrix} = \begin{pmatrix} (-1)(2) + (0)(0) \\ (0)(2) + (1)(0) \end{pmatrix} = \begin{pmatrix} -2 + 0 \\ 0 + 0 \end{pmatrix} = \begin{pmatrix} -2 \\ 0 \end{pmatrix}$$

Kwa Q'(6,-1):

$$Q'' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 6 \\ -1 \end{pmatrix} = \begin{pmatrix} (-1)(6) + (0)(-1) \\ (0)(6) + (1)(-1) \end{pmatrix} = \begin{pmatrix} -6 + 0 \\ 0 - 1 \end{pmatrix} = \begin{pmatrix} -6 \\ -1 \end{pmatrix}$$

Kwa R'(0,3):

$$R'' = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} (-1)(0) + (0)(3) \\ (0)(0) + (1)(3) \end{pmatrix} = \begin{pmatrix} 0 + 0 \\ 0 + 3 \end{pmatrix} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}$$

Vipeo vya picha P''Q''R'' ni:

• P''(-2,0)
• Q''(-6,-1)
• R''(0,3) Chora pointi hizi kwenye gridi na uziunganishe kuunda pembetatu P''Q''R''.

$$T_{total} = T_2 T_1 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix}$$ $$T_{total} = \begin{pmatrix} (-1)(2) + (0)(-1) & (-1)(0) + (0)(1) \\ (0)(2) + (1)(-1) & (0)(0) + (1)(1) \end{pmatrix}$$ $$T_{total} = \begin{pmatrix} -2 + 0 & 0 + 0 \\ 0 - 1 & 0 + 1 \end{pmatrix}$$ $$T_{total} = \begin{pmatrix} -2 & 0 \\ -1 & 1 \end{pmatrix}$$

Kwa matriki $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, kinyume chake ni $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
Kwa $T_{total} = \begin{pmatrix} -2 & 0 \\ -1 & 1 \end{pmatrix}$:
Kiamua (determinant) $ad-bc = (-2)(1) - (0)(-1) = -2 - 0 = -2$.

$$(T_{total})^{-1} = \frac{1}{-2} \begin{pmatrix} 1 & 0 \\ -(-1) & -2 \end{pmatrix}$$ $$(T_{total})^{-1} = \frac{1}{-2} \begin{pmatrix} 1 & 0 \\ 1 & -2 \end{pmatrix}$$ $$(T_{total})^{-1} = \begin{pmatrix} -\frac{1}{2} & 0 \\ -\frac{1}{2} & 1 \end{pmatrix}$$

Matriki inayobadilisha P''Q''R'' kuwa PQR ni $\boxed{\begin{pmatrix} -\frac{1}{2} & 0 \\ -\frac{1}{2} & 1 \end{pmatrix}}$.