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 [latex]A[/latex] is an [latex]m\times n[/latex] rectangular array of elements, arranged in [latex]m[/latex] rows and [latex]n[/latex] columns, denoted
\[ A=(a_{ij})=\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22}\\ \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]. \]
(b) The transpose of [latex]A=(a_{ij})[/latex]} is [latex]A^{T}=(a_{ji}).[/latex]
(c) The conjugate of [latex]A=(a_{ij})[/latex] is [latex]A=(\overline{a_{ij}})[/latex].
(d) A square matrix [latex]A[/latex] has the same number of rows and columns. That is, [latex]A[/latex] is [latex]n\times n[/latex].
(e) A column vector [latex]\overrightarrow{x}[/latex] is an [latex]n\times1[/latex] matrix.
(f) A row vector [latex]\overrightarrow{x}[/latex] is an [latex]1\times n[/latex] matrix.
(g) The zero matrix is defined to be 0 = (0), all entries are zero.
(h) Two matrices [latex]A=(a_{ij})[/latex] and [latex]B=(b_{ij})[/latex] are equal if [latex]a_{ij}=b_{ij}[/latex] for all [latex]i[/latex] and [latex]j[/latex].
Example 1: Find [latex]A^{T}[/latex] and [latex]\overline{A}[/latex] where [latex]A=(a_{ij})=\left[\begin{array}{cc} 2-i & 1-i\\ -3+\sqrt{3}i & 4+2i \end{array}\right].[/latex]
Exercise 1: Find [latex]A^{T}[/latex] and [latex]\overline{A}[/latex] where [latex]A=(a_{ij})=\left[\begin{array}{cc} 4+2i & 3-i\\ -1-3i & 1+I \end{array}\right].[/latex]
Definition: (a) The product of a matrix [latex]A=(a_{ij})[/latex] and a constant [latex]k[/latex] is defined to be [latex]kA=(ka_{ij}).[/latex]
(b) The sum of two [latex]m\times n[/latex] matrices [latex]A=(a_{ij})[/latex] and [latex]B=(b_{ij})[/latex] is defined to be [latex]A+B=(a_{ij}+b_{ij}).[/latex]
(c) The difference of two [latex]m\times n[/latex] matrices [latex]A=(a_{ij})[/latex] and [latex]B=(b_{ij})[/latex] is defined to be [latex]A-B=(a_{ij}-b_{ij}).[/latex]
(d) The product of an [latex]m\times n[/latex] matrix [latex]A=(a_{ij})[/latex] and an [latex]n\times r[/latex] matrix [latex]B=(b_{ij})[/latex] is defined to be the matrix
[latex]C=(c_{ij})[/latex], where
\[c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.\]
Example 2: Find [latex]2A-B[/latex], [latex]\text{AB}[/latex], and [latex]BA[/latex] where [latex]A=(a_{ij})=\left[\begin{array}{cc} 2-i & 1-i\\ -3+3i & 4+2i \end{array}\right][/latex] and [latex]B=(b_{ij})=\left[\begin{array}{cc} 1+i & -2-3i\\ 3-i & 2+4i \end{array}\right][/latex].
Exercise 2: Find [latex]A-3B[/latex], [latex]\text{AB}[/latex], and [latex]BA[/latex] where [latex]A=(a_{ij})=\left[\begin{array}{cc} 4+2i & 3-i\\ -1-3i & 1+I \end{array}\right][/latex] and [latex]B=(b_{ij})=\left[\begin{array}{cc} -1-i & 2+3i\\ 1+i & 1-2i \end{array}\right].[/latex]
Definition: (a) The length of an [latex]1\times n[/latex] vector [latex]\overrightarrow{x}=(x_{1},x_{2},...,x_{n})[/latex] is defined as [latex]\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}}[/latex].
(b) The multiplicative identity matrix [latex]I[/latex] is an [latex]n\times n[/latex] matrix given by
\[ I=\left[\begin{array}{cccc} 1 & 0 & \cdots & 0\\ 0 & 1 & \cdots &0 \\ \vdots & \vdots & \ddots &0 \\ 0 & 0 & \cdots & 1 \end{array}\right]. \]
(c) For any square matrix [latex]A[/latex], it follows that [latex]AI=IA=A[/latex].
(d) A square matrix [latex]A[/latex] is nonsingular, or invertible, if there exists a matrix [latex]B[/latex] such that that [latex]AB=BA=I[/latex].
Otherwise [latex]A[/latex] is singular. The matrix [latex]B[/latex], if it exists, is unique and is denoted by [latex]A^{-1}[/latex] and is called the inverse of [latex]A[/latex].
(e) The determinate of [latex]2\times2[/latex] matrix [latex]A=\left[\begin{array}{cc} a & b\\ c & d \end{array}\right][/latex] is [latex]ad-bc[/latex] which denoted as [latex]\text{det}A=ad-bc[/latex].
(f) The inverse of the [latex]2\times2[/latex] matrix [latex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right][/latex] is
\[\frac{1}{ad-bc}\left[\begin{array}{cc} d & -b\\ -c & a \end{array}\right].\]
Example 3: Find [latex]A^{-1}[/latex] or show it is singular where [latex]A (a_{ij})=\left[\begin{array}{cc} 2-3i & 1-i\\ -3+i & 1+I \end{array}\right][/latex].
Exercise 3: Find [latex]A^{-1}[/latex] or show it is singular where [latex]A=(a_{ij})=\left[\begin{array}{cc} 4-2i & 1-3i\\ 1-i & 1+I \end{array}\right][/latex] .
Group Work
[latex]A=(a_{ij})=\left[\begin{array}{cc} 1-2i & 1+I\\ 3+2i & 4+I \end{array}\right][/latex] and [latex]B=(b_{ij})=\left[\begin{array}{cc} 1+i & 4-3i\\ 2-i & 1+4i \end{array}\right][/latex] .
1. Find [latex]A^{T}[/latex] and [latex]\overline{A}[/latex]
2. [latex]4A-2B[/latex], [latex]\text{AB}[/latex], and [latex]BA[/latex]
3. [latex]A^{-1}[/latex] or show it is singular
4. [latex]B^{-1}[/latex] 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 [latex]x_{1}[/latex] and [latex]x_{2}[/latex]. [latex]x_{1}'=2x_{1}+5x_{2}[/latex], [latex]x_{2}'=x_{1}-2x_{2}[/latex], [latex]x_{1}(0)=0[/latex], [latex]x_{2}(0)=2[/latex].