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Matrix multiplication

Rows times columns, entry by entry.

Entry (i, j) of AB is row i of A dotted with column j of B, which is why the inner dimensions must match. Every entry is computed in the open below. Matrix multiplication is composition of transformations, which is why it is not commutative.

Exemplu lucrat: [[1,2],[3,4]] * [[5,6],[7,8]]

Linear-Algebra · Matrix-Multiplication

[[1,2],[3,4]] * [[5,6],[7,8]]

\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right] \left[\begin{matrix}5 & 6\\7 & 8\end{matrix}\right]

Pas cu pas

  1. \left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right] \left[\begin{matrix}5 & 6\\7 & 8\end{matrix}\right]

    A 2×2 times a 2×2 gives a 2×2 matrix. Entry (i, j) is row i of A dotted with column j of B.

  2. c_{11} = (1)(5) + (2)(7) = 19

  3. c_{12} = (1)(6) + (2)(8) = 22

  4. c_{21} = (3)(5) + (4)(7) = 43

  5. c_{22} = (3)(6) + (4)(8) = 50

  6. AB = \left[\begin{matrix}19 & 22\\43 & 50\end{matrix}\right]

Revelează răspunsul
AB = \left[\begin{matrix}19 & 22\\43 & 50\end{matrix}\right]

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