Mastering Matrix Multiplication: A Step-By-Step Guide With Examples

Product of matrices

Multiplying matrices together.

The product of matrices is a mathematical operation that involves multiplying two matrices to obtain a new matrix. The product is only defined if the number of columns in the first matrix is equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

To find the product of two matrices, we multiply each element in the first row of the first matrix by the corresponding element in the first column of the second matrix, then add up the products. This gives us the first element in the resulting matrix. We repeat this process for all the elements in the first row of the first matrix, then move on to the second row of the first matrix and do the same thing. We continue until we have multiplied all the rows of the first matrix by all the columns of the second matrix.

Here is an example of how to find the product of two matrices:

Suppose we have a 2×3 matrix A and a 3×2 matrix B:

A = [2 3 1]
[1 2 3]

B = [4 5]
[6 7]
[8 9]

To find the product AB, we take the dot product of the first row of A and the first column of B:

(2*4) + (3*6) + (1*8) = 34

This gives us the first element in the resulting matrix. We repeat this process for the other elements in the first row of A and the first column of B:

(2*5) + (3*7) + (1*9) = 32
(1*4) + (2*6) + (3*8) = 32
(1*5) + (2*7) + (3*9) = 50

These values correspond to the elements in the first row of the resulting matrix. We repeat this process for the other rows of A and columns of B to obtain the complete product AB:

AB = [34 32]
[32 50]

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