Matrix Multiplication Calculator

Matrix multiplication relies on the dot product of rows from the first matrix and columns from the second matrix. To multiply two matrices, say Matrix A and Matrix B, the number of columns in Matrix A must strictly equal the number of rows in Matrix B. If Matrix A has dimensions m × n and Matrix B has dimensions n × p, the resulting product Matrix C will have dimensions m × p. Each element $c_{ij}$ in the resulting matrix is computed by taking the sum of the products of corresponding elements from the $i$-th row of Matrix A and the $j$-th column of Matrix B: $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$.

Let us walk through multiplying a 2x2 Matrix A by a 2x2 Matrix B. Suppose Matrix A has elements [[1, 2], [3, 4]] and Matrix B has elements [[5, 6], [7, 8]]. To find the top-left element ($c_{11}$) of the resulting matrix, we multiply the first row of Matrix A ([1, 2]) by the first column of Matrix B ([5, 7]): $(1 \times 5) + (2 \times 7) = 5 + 14 = 19$. Repeating this process for the top-right element ($c_{12}$) yields $(1 \times 6) + (2 \times 8) = 6 + 16 = 22$. For the bottom row, $c_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43$, and $c_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50$. The final product matrix is [[19, 22], [43, 50]].

Always verify your inner dimensions before starting any multiplication; remember that the column count of the left matrix must match the row count of the right matrix. Keep in mind that matrix multiplication is non-commutative, meaning A × B does not generally equal B × A. Double-check your negative signs and scalar placements when entering data into the calculator to ensure absolute accuracy in your final output.