Practical work 1: Matrixes. Operations over matrixes.

Proposition. Multiplication of two matrixes A and B not communicative, that is .

Example. Two matrixes A and B of the third order are given. To find product of matrixes AB and BA and to compare the received results.

Task 1. To find product C=AB of the given matrixes and .

Solution. The number of columns of a matrix A is equal to number of lines of a matrix B, therefore the matrix A can be multiply at the left on a matrix B. By a rule of multiplication of matrixes the element , located on crossing of i -th line and of j -th column of a matrix C, is equal to the sum of products of elements i -th line of matrix A on corresponding elements of j -th column of a matrix B.

Сonsequently we have

.

Task 2. To find a matrix , if .

Solution. The order of the decision of a task:

1. To find product of matrixes А and B.

2. To execute multiplication of number 2 to product АВ.

3. To execute multiplication of number 3 to matrix E.

4. To execute addition of matrixes 2 АВ and 3 Е, that is to find a matrix C.

Answer.

Task 3. The given matrix is . To find the transposed matrix .

Solution. To find the transposed matrix, it is necessary to change in an initial matrix places of a line with columns. Thus, we receive a following matrix

Task 4. The matrix is given. Find the matrix .

Solution. For the decision of the given task it is necessary to multiply a matrix A by itself three times. As a result we will receive a matrix .

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