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.

Haya Terry, twendelee na swali hili la Hisabati. 1. a) Chora PQR yenye vipeo P(1,1), Q(3,2) na R(0,3) kwenye gridi iliyotolewa. Vipeo vya asili ni: P(1,1) Q(3,2) R(0,3) Chora pointi hizi kwenye gridi na uziunganishe kuunda pembetatu PQR. b) Tafuta na uchore picha P'Q'R', picha ya PQR chini ya mabadiliko yenye matriki 2 & 0 \\ -1 & 1 . Step 1: Tumia matriki ya mabadiliko T_1 = 2 & 0 \\ -1 & 1 kupata vipeo vya P'Q'R'. Kwa P(1,1): P' = 2 & 0 \\ -1 & 1 1 \\ 1 = (2)(1) + (0)(1) \\ (-1)(1) + (1)(1) = 2 + 0 \\ -1 + 1 = 2 \\ 0 Kwa Q(3,2): Q' = 2 & 0 \\ -1 & 1 3 \\ 2 = (2)(3) + (0)(2) \\ (-1)(3) + (1)(2) = 6 + 0 \\ -3 + 2 = 6 \\ -1 Kwa R(0,3): R' = 2 & 0 \\ -1 & 1 0 \\ 3 = (2)(0) + (0)(3) \\ (-1)(0) + (1)(3) = 0 + 0 \\ 0 + 3 = 0 \\ 3 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'. c) P''Q''R'' inabadilishwa kuwa P'''Q'''R''' na matriki ya mabadiliko M = -1 & 0 \\ 0 & 1 . Tafuta vipeo vya P'''Q'''R''' na uchore picha. (Kumbuka: Kwa kuzingatia mpangilio wa maswali, tutachukulia kuwa P'Q'R' inabadilishwa kuwa P''Q''R'' na matriki M, na kwamba "P'''Q'''R'''" ni kosa la uchapaji na inapaswa kuwa P''Q''R''.) Step 1: Tumia matriki ya mabadiliko T_2 = M = -1 & 0 \\ 0 & 1 kupata vipeo vya P''Q''R''. Kwa P'(2,0): P'' = -1 & 0 \\ 0 & 1 2 \\ 0 = (-1)(2) + (0)(0) \\ (0)(2) + (1)(0) = -2 + 0 \\ 0 + 0 = -2 \\ 0 Kwa Q'(6,-1): Q'' = -1 & 0 \\ 0 & 1 6 \\ -1 = (-1)(6) + (0)(-1) \\ (0)(6) + (1)(-1) = -6 + 0 \\ 0 - 1 = -6 \\ -1 Kwa R'(0,3): R'' = -1 & 0 \\ 0 & 1 0 \\ 3 = (-1)(0) + (0)(3) \\ (0)(0) + (1)(3) = 0 + 0 \\ 0 + 3 = 0 \\ 3 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''. d) Tafuta matriki inayobadilisha P''Q''R'' kuwa PQR. Hii inamaanisha tunahitaji kupata matriki ya kinyume ya mabadiliko yote kutoka PQR hadi P''Q''R''. Mabadiliko ya jumla kutoka PQR hadi P''Q''R'' yanawakilishwa na matriki T_total = T_2 T_1. Step 1: Kokotoa matriki ya mabadiliko ya jumla T_total. T_total = T_2 T_1 = -1 & 0 \\ 0 & 1 2 & 0 \\ -1 & 1 T_total = (-1)(2) + (0)(-1) & (-1)(0) + (0)(1) \\ (0)(2) + (1)(-1) & (0)(0) + (1)(1) T_total = -2 + 0 & 0 + 0 \\ 0 - 1 & 0 + 1 T_total = -2 & 0 \\ -1 & 1 Step 2: Tafuta kinyume cha matriki T_total. Kwa matriki A = a & b \\ c & d , kinyume chake ni A^-1 = (1)/(ad-bc) d & -b \\ -c & a . Kwa T_total = -2 & 0 \\ -1 & 1 : Kiamua (determinant) ad-bc = (-2)(1) - (0)(-1) = -2 - 0 = -2. (T_total)^-1 = (1)/(-2) 1 & 0 \\ -(-1) & -2 (T_total)^-1 = (1)/(-2) 1 & 0 \\ 1 & -2 (T_total)^-1 = -(1)/(2) & 0 \\ -(1)/(2) & 1 Matriki inayobadilisha P''Q''R'' kuwa PQR ni -(1)/(2) & 0 \\ -(1)/(2) & 1 . Umeelewa vizuri! Tuma swali linalofuata 📸