## dot product of a matrix

### The dot product of two matrices is a mathematical operation that results in a scalar value

The dot product of two matrices is a mathematical operation that results in a scalar value. It is used to determine the similarity or the level of correspondence between two matrices.

The dot product of two matrices, A and B, is calculated by multiplying the corresponding elements of the matrices and summing the products. However, there are specific criteria that need to be met in order to perform the dot product on matrices:

1. The matrices must have compatible dimensions for multiplication. Specifically, the number of columns in matrix A must be equal to the number of rows in matrix B.

2. If matrix A has dimensions m x n, and matrix B has dimensions n x p, the resulting dot product matrix C will have dimensions m x p.

The formula for calculating the dot product of two matrices is as follows:

C[i][j] = Σ (A[i][k] * B[k][j]), for k = 1 to n

where C[i][j] represents the element in the i-th row and j-th column of the resulting matrix C.

To better understand the concept, let’s consider an example:

Suppose we have matrix A = [ 2 3 ] and matrix B = [ 4 1 ].

[ 5 6 ] [ 2 3 ]

For each element in the resulting dot product matrix C, we would multiply and sum:

C[1][1] = (2 * 4) + (3 * 2) = 8 + 6 = 14

C[1][2] = (2 * 1) + (3 * 3) = 2 + 9 = 11

C[2][1] = (5 * 4) + (6 * 2) = 20 + 12 = 32

C[2][2] = (5 * 1) + (6 * 3) = 5 + 18 = 23

Therefore, the resulting dot product matrix C is:

C = [ 14 11 ]

[ 32 23 ]

The dot product of matrices is an important operation in linear algebra and has various applications, such as in solving systems of linear equations, calculating distances between vectors, and determining the angle between vectors.

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