## Matrices operations theory

# Matrices additions and subtractions theory

The sum of two Matrices (Aij), (Bij) of the same dimension, is another matrix (Rij) of the same dimension and with generic term sij=aij+bij.

Therefore, to be able to add two matrices, they must have the same dimension

Addition of Matrices: (Aij) + (Bij) = (Aij + Bij)

Subtraction of Matrices: (Aij) – (Bij) = (Aij – Bij)

Example:

# Matrix multiplications and division theory

For two matrices A and B can be multiplied, A · B, it is necessary that the number of columns in the first matrix be the same as the number of rows in the second one.

In this case, the product A · B = C is another matrix whose elements are obtained by multiplying each row of the first matrix by each column of the second vector, as follows:

A_{ik}·B_{kj}=C_{ij}

The resulting matrix C has many rows as A and many columns as B

Example:

The matrix division is defined as the product of the numerator multiplied by the inverse matrix of the denominator. That is to say:

A/B = A·B^{-1}

To multiply a number by a matrix it is multiplied the number by each matrix digit.

Example:

To divide a matrix by a number all the terms of the matrix are divided by the number.

Example:

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