Matrix Multiplication

Matrix Multiplication

ByAbdul Wahab Junaid

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. The process involves multiplying rows of the first matrix by columns of the second matrix and summing the results. Here’s a detailed explanation:

Preconditions for Matrix Multiplication

Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix.

  • If matrix ( A ) is of size ( m \times n ) (with ( m ) rows and ( n ) columns) and matrix ( B ) is of size ( n \times p ) (with ( n ) rows and ( p ) columns), then their product ( AB ) will be a matrix ( C ) of size ( m \times p ).

How to Multiply Matrices

Given matrices ( A ) and ( B ):

[A = \begin{pmatrix}a_{11} & a_{12} & \dots & a_{1n} \a_{21} & a_{22} & \dots & a_{2n} \\vdots & \vdots & \ddots & \vdots \a_{m1} & a_{m2} & \dots & a_{mn}\end{pmatrix}, \quadB = \begin{pmatrix}b_{11} & b_{12} & \dots & b_{1p} \b_{21} & b_{22} & \dots & b_{2p} \\vdots & \vdots & \ddots & \vdots \b_{n1} & b_{n2} & \dots & b_{np}\end{pmatrix}]

The product matrix ( C = AB ) is defined as:

C=​c11​c21​⋮cm1​​c12​c22​⋮cm2​​……⋱…​c1p​c2p​⋮cmp​​​

Where each element ( c_{ij} ) of the resulting matrix ( C ) is calculated as:

[c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}]

This means that the element ( c_{ij} ) is the dot product of the ( i )-th row of matrix ( A ) and the ( j )-th column of matrix ( B ).

Example of Matrix Multiplication

Let’s multiply two matrices:

[A = \begin{pmatrix}1 & 2 & 3 \4 & 5 & 6\end{pmatrix}, \quadB = \begin{pmatrix}7 & 8 \9 & 10 \11 & 12\end{pmatrix}]

Here, ( A ) is a ( 2 \times 3 ) matrix, and ( B ) is a ( 3 \times 2 ) matrix. The resulting matrix ( C = AB ) will be a ( 2 \times 2 ) matrix.

Let’s calculate each element of matrix ( C ):

  • ( c_{11} ) (first row, first column):
    [c_{11} = (1 \times 7) + (2 \times 9) + (3 \times 11) = 7 + 18 + 33 = 58]
  • [( c_{12} ) (first row, second column)]:
    [c_{12} = (1 \times 8) + (2 \times 10) + (3 \times 12) = 8 + 20 + 36 = 64]
  • ( c_{21} ) (second row, first column):
    [c_{21} = (4 \times 7) + (5 \times 9) + (6 \times 11) = 28 + 45 + 66 = 139]
  • ( c_{22} ) (second row, second column):
    [c_{22} = (4 \times 8) + (5 \times 10) + (6 \times 12) = 32 + 50 + 72 = 154]

So, the resulting matrix ( C ) is:

[C = \begin{pmatrix}58 & 64 \139 & 154\end{pmatrix}]

Properties of Matrix Multiplication

  1. Non-commutative: In general, ( AB \neq BA ). The order in which matrices are multiplied matters.
  2. Associative: Matrix multiplication is associative, meaning ( A(BC) = (AB)C ).
  3. Distributive: Matrix multiplication is distributive over addition, so ( A(B + C) = AB + AC ) and ( (A + B)C = AC + BC ).
  4. Identity Matrix: Multiplying any matrix by the identity matrix ( I ) (where ( I ) is a square matrix with 1’s on the diagonal and 0’s elsewhere) leaves the matrix unchanged: ( AI = IA = A ).
  5. Zero Matrix: Multiplying any matrix by a zero matrix results in a zero matrix.

Matrix multiplication is widely used in various fields, including computer graphics, physics, engineering, economics, and more. It forms the backbone of many algorithms and systems in linear algebra.

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