Solve Combined Matrix Transformation Problems: Find Combined Matrix and Image Coordinate of Point P(3,-2)

Question

Point P(3,-2) is transformed by $\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$, followed by a further transformation $\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix}$. (i) Work out the matrix for the combined transformation. (ii) Work out the x-coordinate of the image point of P.

Accepted Answer

(i) $\begin{bmatrix} -1 & 1 \\ 0 & 1 \end{bmatrix}$; (ii) $-5$

Answer

Step 1: Solve part (i) - Find the combined transformation matrix. When combining transformations, we multiply the matrices in the reverse order of the transformations applied: first apply $\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$, then $\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix}$, so the combined matrix is $\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$. Step 2: Calculate the matrix multiplication. For two 2x2 matrices $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\begin{bmatrix} e & f \\ g & h \end{bmatrix}$, the product is $\begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{bmatrix}$. Applying this: $\begin{bmatrix} 0*1 + 2*0 & 0*(-1) + 2*1 \\ 1*1 + 0*0 & 1*(-1) + 0*1 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}$? Correction: Reverse order is critical, the second transformation matrix is multiplied first with the first transformation matrix: $\begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0*1 + 2*0 & 0*(-1) + 2*1 \\ 1*1 + 0*0 & 1*(-1) + 0*1 \end{bmatrix} = \begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}$? No, wait: first transformation $M1 = \begin{bmatrix}1&-1\\0&1\end{bmatrix}$, second $M2 = \begin{bmatrix}0&2\\1&0\end{bmatrix}$. Combined transformation is $M2*M1$. Calculating each element: Top left: (0*1)+(2*0)=0; Top right: (0*(-1))+(2*1)=2; Bottom left: (1*1)+(0*0)=1; Bottom right: (1*(-1))+(0*1)=-1. So combined matrix is $\begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix}$. Step 3: Solve part (ii) - Find the x-coordinate of the image of P. Represent point P as a column vector $\begin{bmatrix} 3 \\ -2 \end{bmatrix}$. Multiply the combined matrix by this vector: $\begin{bmatrix} 0 & 2 \\ 1 & -1 \end{bmatrix} \times \begin{bmatrix} 3 \\ -2 \end{bmatrix}$. Step 4: Calculate the x-coordinate (top element of the resulting vector): $0*3 + 2*(-2) = 0 - 4 = -4$? Wait, no, correct order: first apply M1 to P, then M2 to the result. First apply M1: $\begin{bmatrix}1&-1\\0&1\end{bmatrix} \times \begin{bmatrix}3\\-2\end{bmatrix} = \begin{bmatrix}1*3 + (-1)*(-2) \\ 0*3 +1*(-2)\end{bmatrix} = \begin{bmatrix}5\\-2\end{bmatrix}$. Then apply M2: $\begin{bmatrix}0&2\\1&0\end{bmatrix} \times \begin{bmatrix}5\\-2\end{bmatrix} = \begin{bmatrix}0*5 +2*(-2) \\1*5 +0*(-2)\end{bmatrix} = \begin{bmatrix}-4\\5\end{bmatrix}$. So x-coordinate is -4. Wait, earlier combined matrix was wrong: combined matrix is M2*M1, which is $\begin{bmatrix}0&2\\1&0\end{bmatrix}*\begin{bmatrix}1&-1\\0&1\end{bmatrix} = \begin{bmatrix}0*1+2*0 & 0*(-1)+2*1 \\1*1+0*0 &1*(-1)+0*1\end{bmatrix}=\begin{bmatrix}0&2\\1&-1\end{bmatrix}$. Then multiply by P: $\begin{bmatrix}0&2\\1&-1\end{bmatrix}*\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}0*3+2*(-2)\\1*3 + (-1)*(-2)\end{bmatrix}=\begin{bmatrix}-4\\5\end{bmatrix}$. So correct combined matrix is $\begin{bmatrix}0&2\\1&-1\end{bmatrix}$, x-coordinate is -4. My earlier mistake was in matrix multiplication order. Correct steps: (i) Combined matrix is M2*M1 = $\begin{bmatrix}0&2\\1&0\end{bmatrix}*\begin{bmatrix}1&-1\\0&1\end{bmatrix}=\begin{bmatrix}0*1+2*0&0*(-1)+2*1\\1*1+0*0&1*(-1)+0*1\end{bmatrix}=\begin{bmatrix}0&2\\1&-1\end{bmatrix}$. (ii) Multiply combined matrix by P's column vector: $\begin{bmatrix}0&2\\1&-1\end{bmatrix}*\begin{bmatrix}3\\-2\end{bmatrix}=\begin{bmatrix}0*3 +2*(-2)\\1*3 + (-1)*(-2)\end{bmatrix}=\begin{bmatrix}-4\\5\end{bmatrix}$. So x-coordinate is -4.