Subsection 2.3.1 Matrix Algebra
Matrix algebra deals with mathematical operations on matrices. Matrix algebra is used to solve problems related to electrical circuits, control systems, structural analysis, quantum mechanics, relativity, input-output models, linear regression, and game theory.
Subsubsection 2.3.1.1 Addition
The addition or subtraction for two \(m\times n\) matrices is defined as \(C=A\pm B\text{.}\) where \(c_{ij}= a_{ij}\pm b_{ij}\) forall \(i\) and \(j\text{,}\) e.g.,
\begin{equation*}
A= {\begin{bmatrix} 3 & 1 & 4\\4 & 0 & 0\\ \end{bmatrix}}
= {\begin{bmatrix} 1 & -1 & 2\\3 & 0 & 1\\ \end{bmatrix}}
+{\begin{bmatrix} 2 & 2 & 2\\1 & 0 & -1\\ \end{bmatrix}}
\end{equation*}
Also, commutative and associative laws are hold good in addition or subtraction of matrices, i.e., \(A+B = B+A\) and \(A+(B+C) = (A+B)+C\text{.}\)
Subsubsection 2.3.1.2 Multiplication
The product of two matrices A and B is only possible if the numbers of column in A is equal to the numbers of rows in B. Let \(A= [a_{ij}]\) be a \(m\times n \) matrix and \(B= [b_{ij}]\) be a \(n\times p\) be matrix, then \(AB= C\) is \(m\times p\) matrix, where \(C= [c_{ij}]= a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj} = \sum\limits_{k=1}^{n}a_{ik}b_{kj}\text{.}\) For example,
\begin{equation*}
{\begin{bmatrix} 0 & 0 \\ 0 & 1 \\ \end{bmatrix}} \cdot {\begin{bmatrix} 0 & 0 \\ 2 & 3 \\ \end{bmatrix}}
={\begin{bmatrix} 0 & 0 \\ 2 & 3 \\ \end{bmatrix}}.
\end{equation*}
Properties of multiplication.
- Matrix multiplication is not commutative i.e., \(AB \neq BA\text{.}\)
- Matrix multiplication is Associative i.e., \(A(BC) =(AB)C\text{.}\)
- Matrix multiplication is Distributive i.e., \(A(B+C) = AB+AC\text{.}\)
- Matrix is multiplicabe by unit matrix i.e., \(AI=IA=A\text{.}\)
- If the non singular matrix is multiplied by its inverse, a unit matrix is produced, i.e.,\begin{equation*} A\cdot A^{-1}=A^{-1}\cdot A = I \end{equation*}(If \(|A| \neq 0\)).
