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Given matrices A = [[1, 2, 3], [4, 3, 6]] and B = [[1, 2], [3, 4], [5, 6]], calculate the product AB.

Mathematics

ScanSolve AI Answer

Answer

[22284356]\begin{bmatrix} 22 & 28 \\ 43 & 56 \end{bmatrix}

Step-by-step solution

5 steps

Given the matrices:

A=[123436]A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 3 & 6 \end{bmatrix} B=[123456]B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}

We need to find the product ABAB.
Matrix A is a 2×32 \times 3 matrix and Matrix B is a 3×23 \times 2 matrix. The resulting matrix ABAB will be a 2×22 \times 2 matrix.

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

    Multiply the first row of A by the first column of B.

    (1×1)+(2×3)+(3×5)=1+6+15=22(1 \times 1) + (2 \times 3) + (3 \times 5) = 1 + 6 + 15 = 22

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

    Multiply the first row of A by the second column of B.

    (1×2)+(2×4)+(3×6)=2+8+18=28(1 \times 2) + (2 \times 4) + (3 \times 6) = 2 + 8 + 18 = 28

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

    Multiply the second row of A by the first column of B.

    (4×1)+(3×3)+(6×5)=4+9+30=43(4 \times 1) + (3 \times 3) + (6 \times 5) = 4 + 9 + 30 = 43

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

    Multiply the second row of A by the second column of B.

    (4×2)+(3×4)+(6×6)=8+12+36=56(4 \times 2) + (3 \times 4) + (6 \times 6) = 8 + 12 + 36 = 56

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

    AB=[22284356]AB = \begin{bmatrix} 22 & 28 \\ 43 & 56 \end{bmatrix}

    The product of the matrices is:

    [22284356]\boxed{\begin{bmatrix} 22 & 28 \\ 43 & 56 \end{bmatrix}}
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