Multiplication of matrices and its application in biology
Multiplication of Matrices and its application in biology
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.
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).
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
Properties of Matrix Multiplication1. Matrix multiplication is not commutative in nature i.e. AB ≠ BA2. It is associative in nature i.e. A(BC) = (AB)C3. It is distributive over matrix multiplication i.e. A(B+C) = AB + BC (A+ C)B = AB + BC
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.
Applications of Matrix Multiplication in Biology1. 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.
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.
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.
• 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.
Bibliography• www.googleimages.com• www.biology301.com• www.wikipedia.com• NCERT maths for class 12th- part I• Engineering maths by V. P. Mishra