Lecture One: Matrices Basic Definitions and Operations

Basic Definitions

• Matrix: A matrix is a rectangular array of numbers (entries) arranged in rows and columns.

• Order: The order of a matrix is given as $m\times n$, where $m$ is the number of rows and $n$ is the number of columns.

• Element: The element in row $i$ and column $j$ of a matrix is denoted as $a_{ij}$ and it's called the $(i,j{)}^{th}$ entry of the matrix $A$.

• Main Diagonal: The main diagonal of any matrix is the set of entries $a_{ij}$ where $i=j$.

Special Types of Matrices

1. Row Matrix: A row matrix has 1 row and $n$ columns. Example:

$$A=\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$$

2. Column Matrix: A column matrix has $m$ rows and 1 column. Example:

$$A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$$

3. Square Matrix: A square matrix has the same number of rows and columns, i.e., $m=n$. Examples:

$$\begin{array}{c}{A}_{2\times 2}=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\\ \\ B_{3\times 3}=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]\end{array}$$

4. Identity Matrix: The identity matrix $I$ is a square matrix where all diagonal elements are 1, and all other elements are 0. Examples:

$$\begin{array}{c}{I}_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\\ \\ I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$$

5. Zero Matrix: The zero matrix $O$ has all its entries as 0. Examples:

$$\begin{array}{c}{A}_{1\times 3}=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]\\ \\ B_{2\times 2}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]\end{array}$$

Matrix Algebra: Addition, Subtraction, and Scalar Multiplication

1. Matrix Addition: The matrix sum $A+B$ is obtained by adding corresponding entries of $A$ and $B$.

   NOTE

   Matrix addition is only possible if the matrices have the same order.

   Example 1: Find $A+B$:

$$\begin{array}{c}A=\left[\begin{array}{cc}0& -3\\ 5& 2\end{array}\right],B=\left[\begin{array}{cc}13& 1\\ -4& 5\end{array}\right]\\ \\ A+B=\left[\begin{array}{cc}0+13& -3+1\\ 5-4& 2+5\end{array}\right]\\ \\ A+B=\left[\begin{array}{cc}13& -2\\ 1& 7\end{array}\right]\end{array}$$

2. Matrix Subtraction: The matrix difference $A-B$ is obtained by subtracting corresponding entries of $B$ from $A$.

   Example 2: Find $P-Q$:

$$\begin{array}{c}P=\left[\begin{array}{ccc}4& -2& 0\\ 1& 0& 37\end{array}\right]\\ \\ Q=\left[\begin{array}{ccc}-4& -2& 2\\ -8& 3& 6\end{array}\right]\\ \\ P-Q=\left[\begin{array}{ccc}4+4& -2+2& 0-2\\ 1+8& 0-3& 37-6\end{array}\right]\\ \\ P-Q=\left[\begin{array}{ccc}8& 0& -2\\ 9& -3& 31\end{array}\right]\end{array}$$

3. Scalar Multiplication: The scalar multiplication of $A$ by $k$ is obtained by multiplying each entry of $A$ by $k$.

   Example 3: Find $\frac{1}{2}A$:

$$\begin{array}{c}A=\left[\begin{array}{ccc}4& -4& 0\\ 12& 6& 2\\ 8& 18& -2\end{array}\right]\\ \\ \frac{1}{2}A=\frac{1}{2}\left[\begin{array}{ccc}4& -4& 0\\ 12& 6& 2\\ 8& 18& -2\end{array}\right]\\ \\ \frac{1}{2}A=\left[\begin{array}{ccc}2& -2& 0\\ 6& 3& 1\\ 4& 9& -1\end{array}\right]\end{array}$$

Matrix Multiplication

1. Matrix Product: The product $AB$ of matrices $A$ and $B$ is defined if the number of columns of $A$ equals the number of rows of $B$. The resulting matrix will have the order $m\times p$.

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Example 4: Find $AB$:

$$\begin{array}{c}A=\left[\begin{array}{ccc}1& 2& 0\\ 8& 0& 1\\ 2& 4& 6\end{array}\right],B=\left[\begin{array}{cc}3& 0\\ 1& 2\\ 3& 4\end{array}\right]\\ \\ AB=\left[\begin{array}{cc}1(3)+2(1)+0(3)& 1(0)+2(2)+0(4)\\ 8(3)+0(1)+1(3)& 8(0)+0(2)+1(4)\\ 2(3)+4(1)+6(3)& 2(0)+4(2)+6(4)\end{array}\right]\\ \\ AB=\left[\begin{array}{cc}3+2+0& 0+4+0\\ 24+0+3& 0+0+4\\ 6+4+18& 0+8+24\end{array}\right]\\ \\ AB=\left[\begin{array}{cc}5& 4\\ 27& 4\\ 28& 32\end{array}\right]\end{array}$$

2. Non-commutativity: In general, matrix multiplication is not commutative, i.e., $AB\ne BA$.

   Example 5: Show that $AB\ne BA$ for the matrices:

$$A=\left[\begin{array}{cc}0& 3\\ 5& 2\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$$

First $AB$,

$$\begin{array}{c}AB=\left[\begin{array}{cc}0(1)+3(0)& 0(0)+3(2)\\ 5(1)+2(0)& 5(0)+2(2)\end{array}\right]\\ \\ AB=\left[\begin{array}{cc}0& 6\\ 5& 4\end{array}\right]\end{array}$$

Next $BA$,

$$\begin{array}{c}BA=\left[\begin{array}{cc}1(0)+0(5)& 1(3)+0(2)\\ 0(0)+2(5)& 0(3)+2(2)\end{array}\right]\\ \\ BA=\left[\begin{array}{cc}0& 3\\ 10& 4\end{array}\right]\end{array}$$

Hence,

$$\left[\begin{array}{cc}0& 6\\ 5& 4\end{array}\right]\ne \left[\begin{array}{cc}0& 3\\ 10& 4\end{array}\right],\therefore AB\ne BA$$

Matrix Transpose

1. Transpose: The transpose $A^T$ of a matrix $A$ is obtained by swapping the rows and columns of $A$. If $A$ is of order $m\times n$, then $A^T$ will be of order $n\times m$.

   Example 6: Find $A^T$:

$$A=\left[\begin{array}{ccc}3& 7& 0\\ 5& 9& 46\end{array}\right],A^T=\left[\begin{array}{cc}3& 5\\ 7& 9\\ 0& 46\end{array}\right]$$

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NOTE

A square matrix $A$ can be multiplied by itself any number of times, giving the $n^{th}$ power of $A$.

$$A^2=A\cdot A,A^3=A\cdot A\cdot A$$

Basic Properties of Matrices

1. Commutativity of Addition:$$A+B=B+A$$
2. Associativity of Addition:$$(A+B)+C=A+(B+C)$$
3. Distributivity of Scalar Multiplication:$$k(A+B)=kA+kB$$
4. Transposition Properties:$$(A+B{)}^T=A^T+B^T$$$$(A^T{)}^T=A$$$$(kA{)}^T=k{A}^T$$
5. Matrix Multiplication:$$A(BC)=(AB)C$$$$A(B+C)=AB+AC$$
6. Transposition of a Product:$$(AB{)}^T=B^T{A}^T$$
7. Power of a Matrix:$$A^m{A}^n=A^{m+n}=A^n{A}^m$$

Important Special Properties

• $A+O=O+A=A$, where $O$ is the zero matrix.
• $AI=1\times A=A$, where $I$ is the identity matrix.
• $AO=0\times A=O$, where $O$ is the zero matrix.

Thus, the identity and zero matrices behave like the numbers 1 and 0 respectively in ordinary arithmetic and algebra.

Example 7

Show that $A^2=pA+q{I}_2$, then write $A^3$ in the form $A^3=gA+h{I}_2$. Find the values of $\text{p, q, g, h}$.

Given:

$$A=\left[\begin{array}{cc}2& 5\\ 1& 16\end{array}\right]$$

First, compute $A^2$:

$$A^2=\left[\begin{array}{cc}2& 5\\ 1& 16\end{array}\right]\left[\begin{array}{cc}2& 5\\ 1& 16\end{array}\right]$$$$A^2=\left[\begin{array}{cc}2(2)+5(1)& 2(5)+5(16)\\ 1(2)+16(1)& 1(5)+16(16)\end{array}\right]$$$$A^2=\left[\begin{array}{cc}9& 90\\ 18& 261\end{array}\right]$$$$pA=p\left[\begin{array}{cc}2& 5\\ 1& 16\end{array}\right],pA=\left[\begin{array}{cc}2p& 5p\\ p& 16p\end{array}\right]$$$$q{I}_2=q\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],q{I}_2=\left[\begin{array}{cc}q& 0\\ 0& q\end{array}\right]$$

Then, solve for $p$ and $q$ using the equation:

$$A^2=pA+q{I}_2$$$$pA+q{I}_2=\left[\begin{array}{cc}2p& 5p\\ p& 16p\end{array}\right]+\left[\begin{array}{cc}q& 0\\ 0& q\end{array}\right]$$$$=\left[\begin{array}{cc}2p+q& 5p\\ p& 16p+q\end{array}\right]$$

Equating the 2 matrices gives,

$$\left[\begin{array}{cc}2p+q& 5p\\ p& 16p+q\end{array}\right]=\left[\begin{array}{cc}9& 90\\ 18& 261\end{array}\right]$$

This immediately gives $p=18$; substituting this into one of the other equations involving $q$ gives:

$$2p+q=9,2(18)+q=9,q=9-36,q=-27$$

Next, compute $A^3$:

$$\begin{array}{c}{A}^3=A\cdot A^2\\ \\ A(pA+qI)\text{ (substitute }{A}^2\text{ with pA+qI)}\\ \\ A(18A-27I)\Rightarrow \text{ (substitute }p\text{ and }q\text{ with 18 and -27)}\\ \\ 18A^2-27A\text{ (substitute }{A}^2\text{ with pA+qI)}\\ \\ 18(pA+qI)-27A\\ \\ 18(18A-27I)-27A\Rightarrow (p=18,q=-27)\\ \\ 324A-486I_2-27A\\ \\ A^3=297A-486I\\ \\ p=18,q=-27\\ \\ g=297,h=-486\end{array}$$