Matrix multiplication is a fundamental operation in linear algebra. It combines two matrices to produce a new matrix. For multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Let's explore this with an example using two 2x2 matrices.
Before we can multiply matrices, we must check compatibility. For matrix A times B to be defined, the number of columns in matrix A must equal the number of rows in matrix B. In our example, matrix A is 2 by 2 and matrix B is 2 by 2. Since A has 2 columns and B has 2 rows, they are compatible. The resulting matrix C will have 2 rows and 2 columns.
Now let's calculate the first element of the result matrix. To find C₁₁, we take the first row of matrix A and the first column of matrix B. We multiply the corresponding elements: 2 times 5 equals 10, and 3 times 2 equals 6. Then we sum these products: 10 plus 6 equals 16. So C₁₁ equals 16.
Now let's calculate the remaining elements. For C₁₂, we multiply row 1 of A by column 2 of B: 2 times 1 plus 3 times 3 equals 11. For C₂₁, we use row 2 of A and column 1 of B: 1 times 5 plus 4 times 2 equals 13. Finally, for C₂₂: 1 times 1 plus 4 times 3 equals 13. Our final result matrix C has elements 16, 11, 13, and 13.
Let's summarize matrix multiplication. First, always check compatibility by ensuring the number of columns in the first matrix equals the number of rows in the second. The resulting matrix has dimensions equal to the rows of the first matrix by the columns of the second. Each element is calculated as the dot product of the corresponding row and column. Remember that order matters - A times B is not equal to B times A. Matrix multiplication has wide applications in computer graphics, physics, economics, and machine learning, making it an essential mathematical operation.