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Given matrices A and B, calculate 3A + 2B.

Mathematics

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Given the matrices:

A=[24362131]A = \begin{bmatrix} 2 & 4 & 3 & 6 \\ 2 & 1 & 3 & 1 \end{bmatrix} B=[11112222]B = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{bmatrix}

We need to find 3A+2B3A + 2B.

  1. Calculate $3A$ by multiplying each element of matrix A by 3.

    3A=3×[24362131]=[3×23×43×33×63×23×13×33×1]3A = 3 \times \begin{bmatrix} 2 & 4 & 3 & 6 \\ 2 & 1 & 3 & 1 \end{bmatrix} = \begin{bmatrix} 3 \times 2 & 3 \times 4 & 3 \times 3 & 3 \times 6 \\ 3 \times 2 & 3 \times 1 & 3 \times 3 & 3 \times 1 \end{bmatrix} 3A=[6129186393]3A = \begin{bmatrix} 6 & 12 & 9 & 18 \\ 6 & 3 & 9 & 3 \end{bmatrix}

  2. Calculate $2B$ by multiplying each element of matrix B by 2.

    2B=2×[11112222]=[2×12×12×12×12×22×22×22×2]2B = 2 \times \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 1 & 2 \times 1 & 2 \times 1 & 2 \times 1 \\ 2 \times 2 & 2 \times 2 & 2 \times 2 & 2 \times 2 \end{bmatrix} 2B=[22224444]2B = \begin{bmatrix} 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \end{bmatrix}

  3. Add the resulting matrices $3A$ and $2B$ by adding their corresponding elements.

    3A+2B=[6129186393]+[22224444]3A + 2B = \begin{bmatrix} 6 & 12 & 9 & 18 \\ 6 & 3 & 9 & 3 \end{bmatrix} + \begin{bmatrix} 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \end{bmatrix} 3A+2B=[6+212+29+218+26+43+49+43+4]3A + 2B = \begin{bmatrix} 6+2 & 12+2 & 9+2 & 18+2 \\ 6+4 & 3+4 & 9+4 & 3+4 \end{bmatrix}

  4. Perform the addition for each element.

    3A+2B=[8141120107137]3A + 2B = \begin{bmatrix} 8 & 14 & 11 & 20 \\ 10 & 7 & 13 & 7 \end{bmatrix}

    The result of 3A+2B3A + 2B is:

    [8141120107137]\boxed{\begin{bmatrix} 8 & 14 & 11 & 20 \\ 10 & 7 & 13 & 7 \end{bmatrix}}
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