Matrix multiplication

• Matrix multiplication
• Properties of matrix multiplication
• Examples of matrix multiplication

c_ij = a_i1 · b_1j + a_i2 · b_2j + ... + a_in · b_nj

• (A · B) · C= A · (B · C) - matrix multiplication is associative;
• (z · A) · B= z · (A · B), where z - number;
• A · (B + C) = A · B + A · C - matrix multiplication is distributive;
• I_n · A_nm = A_nm · I_m= A_nm - matrix multiplication by identity matrix;
• A · B ≠ B · A - in general, the matrix multiplication is not commutative.
• The product of two matrices is a matrix with as many rows as they have left the matrix, and as many columns as they have the right matrix.

Find the matrix C is the product of matrices A =		4	2		and B =		3	1		.
		9	0				-3	4

Solution:

С = A · B =		4	2		·		3	1		=		6	12
		9	0				-3	4				27	9

The elements of the matrix C are calculated as follows:

c₁₁ = a₁₁·b₁₁ + a₁₂·b₂₁ = 4·3 + 2·(-3) = 12 - 6 = 6

c₁₂ = a₁₁·b₁₂ + a₁₂·b₂₂ = 4·1 + 2·4 = 4 + 8 = 12

c₂₁ = a₂₁·b₁₁ + a₂₂·b₂₁ = 9·3 + 0·(-3) = 27 + 0 = 27

c₂₂ = a₂₁·b₁₂ + a₂₂·b₂₂ = 9·1 + 0·4 = 9 + 0 = 9

Find the matrix C is the product of matrices A =

2   1   -3   0   4   -1

and B =

5   -1   6   -3   0   7

.

Solution:

C = A · B =

2   1   -3   0   4   -1

·

5   -1   6   -3   0   7

=

7   -2   19   -15   3   -18   23   -4   17

The elements of the matrix C are calculated as follows:

c₁₁ = a₁₁·b₁₁ + a₁₂·b₂₁ = 2·5 + 1·(-3) = 10 - 3 = 7

c₁₂ = a₁₁·b₁₂ + a₁₂·b₂₂ = 2·(-1) + 1·0 = -2 + 0 = -2

c₁₃ = a₁₁·b₁₃ + a₁₂·b₂₃ = 2·6 + 1·7 = 12 + 7 = 19

c₂₁ = a₂₁·b₁₁ + a₂₂·b₂₁ = (-3)·5 + 0·(-3) = -15 + 0 = -15

c₂₂ = a₂₁·b₁₂ + a₂₂·b₂₂ = (-3)·(-1) + 0·0 = 3 + 0 = 3

c₂₃ = a₂₁·b₁₃ + a₂₂·b₂₃ = (-3)·6 + 0·7 = -18 + 0 = -18

c₃₁ = a₃₁·b₁₁ + a₃₂·b₂₁ = 4·5 + (-1)·(-3) = 20 + 3 = 23

c₃₂ = a₃₁·b₁₂ + a₂₂·b₂₂ = (4)·(-1) + (-1)·0 = -4 + 0 = -4

c₃₃ = a₃₁·b₁₃ + a₃₂·b₂₃ = 4·6 + (-1)·7 = 24 - 7 = 17