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Matrix multiplication is a mathematical operation that takes two matrices as input and produces a new matrix as output. It involves multiplying each element of one matrix by the corresponding elements in the other matrix and then summing the products. The resulting matrix will have dimensions that depend on the dimensions of the input matrices.
The purpose of matrix multiplication is to combine and transform data from one matrix into another matrix. It is commonly used in various fields of science and engineering to solve complex problems and perform calculations involving large sets of data. It also has applications in computer graphics, data compression, and machine learning.
Matrix multiplication is different from regular multiplication in several ways. First, it involves multiplying two matrices, rather than two single numbers. Second, the order of multiplication matters in matrix multiplication, whereas it does not matter in regular multiplication. Lastly, the resulting matrix in matrix multiplication will have different dimensions than the input matrices, whereas in regular multiplication, the product will have the same dimensions as the inputs.
There are several rules that govern matrix multiplication. First, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Second, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. Lastly, the order of multiplication matters and the product of two matrices will be different depending on the order they are multiplied in.
Matrix multiplication has many real-world applications, especially in fields that deal with large datasets and complex calculations. It is used in physics to solve systems of equations, in computer science for data compression and image processing, in economics for input-output analysis, and in biology to model population dynamics. It is also used in machine learning and artificial intelligence algorithms to process and analyze large amounts of data.