Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. This operation is essential in various fields such as physics, engineering, computer science, and economics.

Key Properties of Matrix Multiplication

 

Non-Commutative: AB≠BAAB neq BAAB=BA in general.

 

Associative: A(BC)=(AB)CA(BC) = (AB)CA(BC)=(AB)C.

 

Distributive: A(B+C)=AB+ACA(B + C) = AB + ACA(B+C)=AB+AC.

 

Multiplication of Matrices

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix will have the number of rows of the first matrix and the number of columns of the second matrix.

Multiplication of Matrices 2x2

Consider two 2x2 matrices: A=(abcd),B=(efgh)A = begin{pmatrix} a & b c & d end{pmatrix}, quad B = begin{pmatrix} e & f g & h end{pmatrix}A=(ac​bd​),B=(eg​fh​)

To find the product C=ABC = ABC=AB: C=(ae+bgaf+bhce+dgcf+dh)C = begin{pmatrix} ae + bg & af + bh ce + dg & cf + dh end{pmatrix}C=(ae+bgce+dg​af+bhcf+dh​)

Step-by-Step Process:

Calculate the element in the first row, first column (C11): C11=ae+bgC_{11} = ae + bgC11​=ae+bg

Calculate the element in the first row, second column (C12): C12=af+bhC_{12} = af + bhC12​=af+bh

Calculate the element in the second row, first column (C21): C21=ce+dgC_{21} = ce + dgC21​=ce+dg

Calculate the element in the second row, second column (C22): C22=cf+dhC_{22} = cf + dhC22​=cf+dh

The resulting matrix CCC is: C=(ae+bgaf+bhce+dgcf+dh)C = begin{pmatrix} ae + bg & af + bh ce + dg & cf + dh end{pmatrix}C=(ae+bgce+dg​af+bhcf+dh​)

Detailed Example

Let's consider specific values for AAA and BBB:

A=(1234),B=(5678)A = begin{pmatrix} 1 & 2 3 & 4 end{pmatrix}, quad B = begin{pmatrix} 5 & 6 7 & 8 end{pmatrix}A=(13​24​),B=(57​68​)

Using the steps outlined above:

C11=1⋅5+2⋅7=5+14=19C_{11} = 1 cdot 5 + 2 cdot 7 = 5 + 14 = 19C11​=1⋅5+2⋅7=5+14=19

C12=1⋅6+2⋅8=6+16=22C_{12} = 1 cdot 6 + 2 cdot 8 = 6 + 16 = 22C12​=1⋅6+2⋅8=6+16=22

C21=3⋅5+4⋅7=15+28=43C_{21} = 3 cdot 5 + 4 cdot 7 = 15 + 28 = 43C21​=3⋅5+4⋅7=15+28=43

C22=3⋅6+4⋅8=18+32=50C_{22} = 3 cdot 6 + 4 cdot 8 = 18 + 32 = 50C22​=3⋅6+4⋅8=18+32=50

The product matrix CCC is: C=(19224350)C = begin{pmatrix} 19 & 22 43 & 50 end{pmatrix}C=(1943​2250​)

Summary

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second matrix. For 2x2 matrices, this results in a new 2x2 matrix where each element is computed by summing the products of corresponding elements from the rows and columns. This process is widely applicable and forms the basis for many advanced mathematical and computational techniques.