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Consider this matrix transformation: [[-1,-2],[4,-1]] What is the image of [[3],[-2]] under this transformation?

Consider this matrix transformation:\newline[−1−24−1] \left[\begin{array}{cc} -1 & -2 \\ & \\ 4 & -1 \end{array}\right] \newlineWhat is the image of [3−2] \left[\begin{array}{c}3 \\ -2\end{array}\right] under this transformation?

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Q. Consider this matrix transformation:\newline[−1−24−1] \left[\begin{array}{cc} -1 & -2 \\ & \\ 4 & -1 \end{array}\right] \newlineWhat is the image of [3−2] \left[\begin{array}{c}3 \\ -2\end{array}\right] under this transformation?
  1. Write Matrix and Vector: Write down the matrix and the vector to be transformed.\newlineMatrix: [−1−2 4−1]\begin{bmatrix} -1 & -2 \ 4 & -1 \end{bmatrix}\newlineVector: [3 −2]\begin{bmatrix} 3 \ -2 \end{bmatrix}\newlineWe will multiply the matrix by the vector to find the image.
  2. Perform Matrix Multiplication: Perform the matrix multiplication.\newlineFirst row of the matrix multiplied by the vector:\newline(−1×3)+(−2×−2)=−3+4=1(-1 \times 3) + (-2 \times -2) = -3 + 4 = 1\newlineSecond row of the matrix multiplied by the vector:\newline(4×3)+(−1×−2)=12+2=14(4 \times 3) + (-1 \times -2) = 12 + 2 = 14\newlineThe resulting vector is:\newline\begin{bmatrix}1\14\end{bmatrix}
  3. Verify Calculations: Verify the calculations to ensure there are no math errors.\newlineFirst element: (−1×3)+(−2×−2)=−3+4=1(-1 \times 3) + (-2 \times -2) = -3 + 4 = 1 (Correct)\newlineSecond element: (4×3)+(−1×−2)=12+2=14(4 \times 3) + (-1 \times -2) = 12 + 2 = 14 (Correct)\newlineThe calculations are correct.

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