2. Introduction
• Matrices are rectangular arrays with entries from an
arbitrary field. An m × n (read "m by n") matrix is thus an array (aik)
where i changes from 1 through m whereas k ranges
from 1 through n. More explicitly,
a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
a31 a32 a33 ... a3n
...
am1 am2 am3 ... amn
which has m rows and n columns.
• In mathematics, matrix multiplication is a binary operation that
takes a pair of matrices, and produces another matrix.
• Note: Multiplication of two matrices can only be carried out if the
number of columns of the first matrix is equal to the number of
rows of the second matrix.
3. How to carry out multiplication
in matrices
1. Consider two matrices A and B, as
shown in the figure. Let the product
matrix be C.
2. We can obtain the first term of the
matrix by the following operation:
C1= (A1 x B1) + (A2
x B3).
3. We can obtain the other terms of the
product matrix in a similar manner.
C2 = (A1 x B2) + (A2 x B4)
C3 = (A3 x B1) + (A4 x B3)
C4 = (A3 x B2) + (A4 x B4)
4. In other cases, if the first matrix is of the
order (m x n) and the second matrix is of
the order (n x p), then the resultant
matrix will be of the order (m x p).
4. Example
• Find out the product of the following matrix:
A= 1 2 3 B= 0 1 2
1 0 0 4 6 0
4 1 2 1 0 3
The first term will equal to
= (1 x 0) + (2 x 4) + (3 x 1)
=8+3
= 11
5. Properties of Matrix
Multiplication
1. Matrix multiplication is not commutative in
nature i.e. AB ≠ BA
2. It is associative in nature
i.e. A(BC) = (AB)C
3. It is distributive over matrix multiplication
i.e. A(B+C) = AB + BC
(A+ C)B = AB + BC
6. 4. Scalar multiplication commutes with matrix
multiplication.
λ(AB) = (λA)B
(AB)λ = A(Bλ)
where λ is a scalar.
5. For matrix transpose
(AB)T = BTAT
where T denotes transpose( the interchanging of
row i with column i in a matrix).
6. For a square matrix A
AI = IA = A
where I is an identity matrix of same
matrix.
7. Applications of Matrix Multiplication
in Biology
1. Due to recent progress of DNA microarray
technology, a large number of gene
expression profile data are being produced.
Matrix multiplication is used to analyze gene
expression in computational molecular
biology. Matrix multiplication is used in this
technology to create simple algorithms.
8. 2. In the circulatory system,
the red blood cells are
constantly being
destroyed and replaced.
Since they carry oxygen
throughout the body,
their numbers are fixed.
We can use matrix
multiplication to find out
the level of red blood
corpuscles in a person.
9. 3. Matrix multiplication is used to
find the frequency of sickle cell
allele of the gene for hemoglobin
causes red blood cells to
collapse. Individuals with
heterozygous genes for sickle cell
develop immunity against
malaria, but individuals
homozygous for this gene tend
to die at an early age. Matrix
multiplication can be used to
find out the frequency of
occurrence of this gene in
individuals living in a certain
area and to calculate the
possibility of this disease
happening in a progeny of a
family.
10. • Human populations have
been increasing at a nearly
exponential rate over the last
couple of thousand years.
Matrix multiplication is used
for calculating population
expansion of a species, not
just human beings, over a
period of time, provided it
grows at a constant rate. This
can help monitor the
population of an endangered
or over-populated species.
11. Bibliography
• www.googleimages.com
• www.biology301.com
• www.wikipedia.com
• NCERT maths for class 12th- part I
• Engineering maths by V. P. Mishra
