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Calculate the product of matrices X and Y: X = [[1, 2], [3, 4]], Y = [[5, 6], [7, 8]]

Mathematics

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Answer

d

c. [19224350]\text{c. } \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

Step-by-step solution

5 steps

To find the product of matrices XX and YY, we perform matrix multiplication.

Given matrices:

X=[1234]X = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} Y=[5678]Y = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

  1. Calculate the element in the first row, first column of XYXY.

    Multiply the first row of XX by the first column of YY:

    (1×5)+(2×7)=5+14=19(1 \times 5) + (2 \times 7) = 5 + 14 = 19

  2. Calculate the element in the first row, second column of XYXY.

    Multiply the first row of XX by the second column of YY:

    (1×6)+(2×8)=6+16=22(1 \times 6) + (2 \times 8) = 6 + 16 = 22

  3. Calculate the element in the second row, first column of XYXY.

    Multiply the second row of XX by the first column of YY:

    (3×5)+(4×7)=15+28=43(3 \times 5) + (4 \times 7) = 15 + 28 = 43

  4. Calculate the element in the second row, second column of XYXY.

    Multiply the second row of XX by the second column of YY:

    (3×6)+(4×8)=18+32=50(3 \times 6) + (4 \times 8) = 18 + 32 = 50

  5. Combine the calculated elements to form the product matrix XYXY.

    XY=[19224350]XY = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

    Comparing this result with the given options:
    a. [19261118]\begin{bmatrix} 19 & 26 \\ 11 & 18 \end{bmatrix}
    b. [11181522]\begin{bmatrix} 11 & 18 \\ 15 & 22 \end{bmatrix}
    c. [19224350]\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}
    d. [11141518]\begin{bmatrix} 11 & 14 \\ 15 & 18 \end{bmatrix}

    The calculated product matches option c.

    The final answer is c. [19224350]\boxed{\text{c. } \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}}.

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