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Given matrices S = [[1, 2, 3], [4, 5, 6]] and Q = [[-1, 4], [0, 1], [2, 7]], compute S + Q^T.

Mathematics

ScanSolve AI Answer

Step-by-step solution

4 steps
  1. Identify the given matrices SS and QQ.

    S=(123456)S = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} Q=(−140127)Q = \begin{pmatrix} -1 & 4 \\ 0 & 1 \\ 2 & 7 \end{pmatrix}
  2. Compute the transpose of matrix QQ, denoted as QTQ^T.

    To find the transpose, swap the rows and columns of QQ.
    The first row of QQ becomes the first column of QTQ^T.
    The second row of QQ becomes the second column of QTQ^T.
    The third row of QQ becomes the third column of QTQ^T.

    QT=(−102417)Q^T = \begin{pmatrix} -1 & 0 & 2 \\ 4 & 1 & 7 \end{pmatrix}
  3. Compute the sum S+QTS + Q^T.

    To add matrices, add their corresponding elements. Both SS and QTQ^T are 2×32 \times 3 matrices, so they can be added.

    S+QT=(123456)+(−102417)S + Q^T = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} + \begin{pmatrix} -1 & 0 & 2 \\ 4 & 1 & 7 \end{pmatrix} S+QT=(1+(−1)2+03+24+45+16+7)S + Q^T = \begin{pmatrix} 1 + (-1) & 2 + 0 & 3 + 2 \\ 4 + 4 & 5 + 1 & 6 + 7 \end{pmatrix} S+QT=(0258613)S + Q^T = \begin{pmatrix} 0 & 2 & 5 \\ 8 & 6 & 13 \end{pmatrix}
  4. Compare the result with the given options.

    The calculated matrix is (0258613)\begin{pmatrix} 0 & 2 & 5 \\ 8 & 6 & 13 \end{pmatrix}. This matches option A.

    The final answer is A. (0258613)\boxed{\text{A. } \begin{pmatrix} 0 & 2 & 5 \\ 8 & 6 & 13 \end{pmatrix}}.

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