Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a new matrix. The key requirement is that the number of columns in the first matrix must equal the number of rows in the second matrix. Here we see two 2 by 2 matrices being multiplied to produce another 2 by 2 matrix.
Let's work through a concrete example step by step. We have matrix A with elements 2, 3, 1, 4 and matrix B with elements 5, 6, 7, 8. To calculate c one one, we multiply row 1 of A with column 1 of B: 2 times 5 plus 3 times 7 equals 31. For c one two, we multiply row 1 of A with column 2 of B: 2 times 6 plus 3 times 8 equals 36. Similarly, c two one equals 1 times 5 plus 4 times 7 equals 33, and c two two equals 1 times 6 plus 4 times 8 equals 38. The final result matrix C has elements 31, 36, 33, and 38.
Understanding matrix dimensions is crucial for multiplication. For matrix A with dimensions m by n to multiply with matrix B with dimensions p by q, the number of columns in A, which is n, must equal the number of rows in B, which is p. For example, a 2 by 3 matrix can multiply with a 3 by 4 matrix, resulting in a 2 by 4 matrix. However, a 2 by 3 matrix cannot multiply with a 2 by 4 matrix because 3 does not equal 2. This compatibility rule is fundamental to matrix multiplication.
Matrix multiplication has several important properties. First, it is non-commutative, meaning AB does not equal BA in general. For example, when we multiply our matrices A and B, we get different results depending on the order. AB gives us the matrix with elements 19, 22, 43, 50, while BA gives us 23, 34, 31, 46. Second, matrix multiplication is associative, so the order of grouping doesn't matter. Third, it's distributive over addition. Finally, there exists an identity matrix I, where multiplying any matrix by I gives the original matrix back.
To summarize what we have learned about matrix multiplication: First, matrix multiplication requires compatible dimensions where the number of columns in the first matrix equals the number of rows in the second matrix. Second, each element in the result matrix is calculated using dot products of corresponding rows and columns. Third, the operation is non-commutative, meaning order matters, but it is associative. Finally, matrix multiplication is essential for linear transformations and solving systems of equations in mathematics, physics, and engineering.