Section 5.2. Introduction of 2 by 2 matrices

Objective:

1. Basic terms of a 2 by 2 matrix. Transpose, Conjugate, Square Matrices, Vectors, The Zero Matrix, Matrix Equality.

2. Basic operations of a 2 by 2 matrix. Scalar Multiplication, Matrix Addition and Subtraction, Matrix Multiplication, Vector Length, Identity Matrix, Determinate of a 2 by 2 matrix, Inverse Matrix.

In this section, we are learning all the definitions and operations that we need for solving a system of ODE. 

Definition: (a) A matrix $A$ is an $m\times n$ rectangular array of elements, arranged in $m$ rows and $n$ columns, denoted 

$$A=(a_{ij})=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}\\ ⋮& ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right].$$

(b) The transpose of $A=(a_{ij})$} is $A^T=(a_{ji}).$

(c) The conjugate of $A=(a_{ij})$ is $A=(\overline{a_{ij}})$.

(d) A square matrix $A$ has the same number of rows and columns. That is, $A$ is $n\times n$. 

(e) A column vector $\overrightarrow{x}$ is an $n\times 1$ matrix.

(f) A row vector [latex]\overrightarrow{x}[/latex] is an $1\times n$ matrix.

(g) The zero matrix is defined to be 0 = (0), all entries are zero.

(h) Two matrices $A=(a_{ij})$ and $B=(b_{ij})$ are equal if $a_{ij}=b_{ij}$ for all $i$ and $j$. 

Example 1: Find $A^T$ and $\bar{A}$ where $A=(a_{ij})=\left[\begin{array}{cc}2-i& 1-i\\ -3+\sqrt{3}i& 4+2i\end{array}\right].$

Exercise 1: Find $A^T$ and $\bar{A}$ where $A=(a_{ij})=\left[\begin{array}{cc}4+2i& 3-i\\ -1-3i& 1+i\end{array}\right].$

Definition: (a) The product of a matrix $A=(a_{ij})$ and a constant $k$ is defined to be $kA=(k{a}_{ij}).$

(b) The  sum of two $m\times n$ matrices $A=(a_{ij})$ and $B=(b_{ij})$ is defined to be $A+B=(a_{ij}+b_{ij}).$

(c) The difference of two $m\times n$ matrices $A=(a_{ij})$ and $B=(b_{ij})$ is defined to be $A-B=(a_{ij}-b_{ij}).$

(d) The product of an $m\times n$ matrix $A=(a_{ij})$ and an $n\times r$ matrix $B=(b_{ij})$ is defined to be the matrix

$C=(c_{ij})$, where 

$$c_{ij}=\sum _{k=1}^n{a}_{ik}{b}_{kj}.$$

Example 2: Find $2A-B$, $\text{AB}$, and $BA$ where $A=(a_{ij})=\left[\begin{array}{cc}2-i& 1-i\\ -3+3i& 4+2i\end{array}\right]$ and $B=(b_{ij})=\left[\begin{array}{cc}1+i& -2-3i\\ 3-i& 2+4i\end{array}\right]$.

Exercise 2: Find $A-3B$, $\text{AB}$, and $BA$ where $A=(a_{ij})=\left[\begin{array}{cc}4+2i& 3-i\\ -1-3i& 1+i\end{array}\right]$ and $B=(b_{ij})=\left[\begin{array}{cc}-1-i& 2+3i\\ 1+i& 1-2i\end{array}\right].$

Definition: (a) The length of an $1\times n$ vector $\overrightarrow{x}=(x_1,x_2,...,x_n)$ is defined as $\sqrt{x_1^2+x_2^2+...+x_n^2}$.

(b) The multiplicative identity matrix $I$ is an $n\times n$ matrix given by

$$I=\left[\begin{array}{cccc}1& 0& \cdots & 0\\ 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & 0\\ 0& 0& \cdots & 1\end{array}\right].$$

(c) For any square matrix $A$, it follows that $AI=IA=A$.

(d) A square matrix $A$ is nonsingular, or invertible, if there exists a matrix $B$ such that that $AB=BA=I$.

Otherwise $A$ is singular. The matrix $B$, if it exists, is unique and is denoted by $A^{-1}$ and is called the inverse of $A$.

(e) The determinate of $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $ad-bc$ which denoted as $\text{det}A=ad-bc$.

(f) The inverse of the $2\times 2$ matrix $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is 

$$\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right].$$

Example 3: Find $A^{-1}$ or show it is singular where $A(a_{ij})=\left[\begin{array}{cc}2-3i& 1-i\\ -3+i& 1+i\end{array}\right]$.

Exercise 3: Find $A^{-1}$ or show it is singular where $A=(a_{ij})=\left[\begin{array}{cc}4-2i& 1-3i\\ 1-i& 1+i\end{array}\right]$ .

Group Work

$A=(a_{ij})=\left[\begin{array}{cc}1-2i& 1+i\\ 3+2i& 4+i\end{array}\right]$ and $B=(b_{ij})=\left[\begin{array}{cc}1+i& 4-3i\\ 2-i& 1+4i\end{array}\right]$ .

1. Find $A^T$ and $\bar{A}$

2. $4A-2B$, $\text{AB}$, and $BA$

3. $A^{-1}$ or show it is singular

4. $B^{-1}$ or show it is singular

5. Transform the system of first order equations into one 2nd order IVP. Then solve the IVP, and use it to find $x_1$ and $x_2$. $x_1^{\prime }=2x_1+5x_2$, $x_2^{\prime }=x_1-2x_2$, $x_1(0)=0$, $x_2(0)=2$.