Title: Matrix Operations

The sum of matrices

Denote the sum of two matrices and (of the same dimensions) by . The sum is defined by adding entries with the same indices

over all i and j.

Example:

Matrix addition is therefore both commutative and associative.

Scalar multiplication

To multiply a matrix with a real number, we multiply each element with this number.

Properties

r(A+B) = rA + rB

(r+s)A = rA + sA

(rs)A = r(sA)

(A + B)T = AT  + BT

(rA)T = r. AT

Multiplication of a row matrix by a column matrix

This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix). To multiply the row by the column, one multiplies corresponding elements, then adds the results.

Example:

Matrix Multiplication

When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed.

Here is an example of matrix multiplication for two 2x2 matrices

Here is an example of matrices multiplication for a 3x3 matrix

Properties of multiplication of matrices

Associativity

If the multiplication is defined then  holds for all matrices A,B and C.

Distributivity

If the multiplication is defined then and holds for all matrices A,B and C.

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