CIE Amath Summary 3

Chapter 12: MATRICES

Introduction

Matrices are rectangular arrays of numbers that provide a powerful framework for representing and solving systems of linear equations, performing linear transformations, and organizing data. This summary covers matrix structure, operations, multiplication, inverses, and solving simultaneous equations.

Matrix Structure

Definition

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix (read "m by n").

$$ A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$

Notation

• $a_{ij}$ denotes the element in the $i$th row and $j$th column.
• Matrices are often denoted by capital letters: $A$, $B$, $C$, etc.
• The order (or dimension) of a matrix is $m \times n$.

Special Types of Matrices

Type	Description
Row matrix	$1 \times n$ (single row)
Column matrix	$m \times 1$ (single column)
Square matrix	$m=n$ (same number of rows and columns)
Zero matrix ($0$)	All entries are zero
Identity matrix ($I_n$)	Square matrix with $1$s on diagonal, $0$s elsewhere
Diagonal matrix	Non-zero entries only on main diagonal
Triangular matrix	All entries above or below diagonal are zero

Examples

1. $A= \begin{pmatrix} 2&-1&0\\ 3&4&5 \end{pmatrix}$ is a $2\times3$ matrix.
2. $B= \begin{pmatrix} 1&2\\ 3&4 \end{pmatrix}$ is a $2\times2$ square matrix.
3. $I_3= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}$ is the $3\times3$ identity matrix.
4. $C= \begin{pmatrix} 4&0&0\\ 0&-2&0\\ 0&0&1 \end{pmatrix}$ is a diagonal matrix.
5. $D= \begin{pmatrix} 1&2&3\\ 0&4&5\\ 0&0&6 \end{pmatrix}$ is an upper triangular matrix.

Matrix Operations and Definitions

Equality of Matrices

Two matrices $A$ and $B$ are equal if and only if:

1. They have the same order ($m\times n$).
2. Corresponding entries are equal: $a_{ij}=b_{ij}$ for all $i,j$.

Addition and Subtraction

Matrices can be added or subtracted only if they have the same order. Addition is performed element-wise:

$$ (A+B)_{ij}=a_{ij}+b_{ij} $$

Similarly, $$ (A-B)_{ij}=a_{ij}-b_{ij} $$

Properties of Matrix Addition

• Commutative: $A+B=B+A$
• Associative: $(A+B)+C=A+(B+C)$
• Identity: $A+0=A$
• Inverse: $A+(-A)=0$

Scalar Multiplication

Multiplying a matrix by a scalar $k$ multiplies every entry:

$$ (kA)_{ij}=k\cdot a_{ij} $$

Examples

1. Let $$ A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, B=\begin{pmatrix}5&6\\7&8\end{pmatrix} $$
   $$ A+B= \begin{pmatrix} 6&8\\ 10&12 \end{pmatrix} $$
   $$ A-B= \begin{pmatrix} -4&-4\\ -4&-4 \end{pmatrix} $$
   $$ 3A= \begin{pmatrix} 3&6\\ 9&12 \end{pmatrix} $$
2. If $$ A=\begin{pmatrix}1&0\\-1&2\end{pmatrix}, B=\begin{pmatrix}2&1\\0&3\end{pmatrix} $$ Find $2A-B$. $$ 2A= \begin{pmatrix} 2&0\\ -2&4 \end{pmatrix} $$ $$ 2A-B= \begin{pmatrix} 0&-1\\ -2&1 \end{pmatrix} $$

Matrix Multiplication

Condition for Multiplication

Two matrices $A$ ($m\times n$) and $B$ ($p\times q$) can be multiplied if and only if $n=p$. The product $AB$ has order $m\times q$.

Definition

If $A$ is $m\times n$ and $B$ is $n\times p$, then $C=AB$ is $m\times p$ with entries:

$$ c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj} $$

$c_{ij}$ is the dot product of the $i$th row of $A$ and the $j$th column of $B$.

Properties of Matrix Multiplication

• Not commutative: $AB\neq BA$
• Associative: $(AB)C=A(BC)$
• Distributive: $A(B+C)=AB+AC$
• Identity: $AI_n=A$
• Zero: $A\cdot0=0$

Examples

1. $$ A= \begin{pmatrix} 1&2\\ 3&4 \end{pmatrix}, B= \begin{pmatrix} 2&0\\ 1&3 \end{pmatrix} $$ $$ AB= \begin{pmatrix} 4&6\\ 10&12 \end{pmatrix} $$
2. $$ A= \begin{pmatrix} 1&2&3\\ 4&5&6 \end{pmatrix}, B= \begin{pmatrix} 7&8\\ 9&10\\ 11&12 \end{pmatrix} $$ $$ AB= \begin{pmatrix} 58&64\\ 139&154 \end{pmatrix} $$
3. $$ A= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, B= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix} $$ $$ AB= \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix} $$ $$ BA= \begin{pmatrix} 0&0\\ 0&1 \end{pmatrix} $$ Therefore, $$AB\neq BA$$

The Inverse of a $2 \times 2$ Matrix

Definition

For a square matrix $A$, if there exists a matrix $A^{-1}$ such that:

$$ AA^{-1}=A^{-1}A=I $$

then $A^{-1}$ is the inverse of $A$. A matrix that has an inverse is called invertible (or non-singular).

Inverse of a $2 \times 2$ Matrix

For $$ A= \begin{pmatrix} a&b\\ c&d \end{pmatrix} $$

$$ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d&-b\\ -c&a \end{pmatrix} $$

provided that $\det(A)=ad-bc\neq0$.

The Determinant

The quantity $\det(A)=ad-bc$ is called the determinant of the $2\times2$ matrix $A$.

• If $\det(A)\neq0$, $A$ is invertible.
• If $\det(A)=0$, $A$ is singular (no inverse).

Properties of Inverses

• $(A^{-1})^{-1}=A$
• $(AB)^{-1}=B^{-1}A^{-1}$
• $(kA)^{-1}=\frac{1}{k}A^{-1}$
• $\det(A^{-1})=\frac{1}{\det(A)}$

Examples

1. Find the inverse of $$ A= \begin{pmatrix} 2&3\\ 1&4 \end{pmatrix} $$

   $$ \det(A)=2\cdot4-3\cdot1=8-3=5\neq0 $$

   $$ A^{-1} = \frac15 \begin{pmatrix} 4&-3\\ -1&2 \end{pmatrix} $$

   $$ = \begin{pmatrix} \frac45&-\frac35\\ -\frac15&\frac25 \end{pmatrix} $$

   Check: $$ AA^{-1} = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} $$

2. Find the inverse of $$ B= \begin{pmatrix} 1&2\\ 2&4 \end{pmatrix} $$

   $$ \det(B)=1\cdot4-2\cdot2=0 $$

   $B$ is singular, so no inverse exists.

3. Find the inverse of $$ C= \begin{pmatrix} 3&-1\\ -2&1 \end{pmatrix} $$

   $$ \det(C)=3\cdot1-(-1)(-2)=1 $$

   $$ C^{-1} = \begin{pmatrix} 1&1\\ 2&3 \end{pmatrix} $$

4. If $$ A= \begin{pmatrix} 2&5\\ 1&3 \end{pmatrix} $$ verify that $$ A^{-1} = \begin{pmatrix} 3&-5\\ -1&2 \end{pmatrix} $$

   $$ AA^{-1} = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} $$

Simultaneous Linear Equations

Matrix Representation

A system of linear equations can be written as:

$$ AX=B $$

where:

$$ A= \begin{pmatrix} a_{11}&a_{12}&\cdots&a_{1n}\\ \vdots&&&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn} \end{pmatrix} $$

$$ X= \begin{pmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{pmatrix} $$

and $$ B= \begin{pmatrix} b_1\\ b_2\\ \vdots\\ b_m \end{pmatrix} $$

Solving $2\times2$ Systems Using Inverse

For an invertible matrix $A$:

$$ X=A^{-1}B $$

Method

1. Write the system in matrix form $AX=B$.
2. Find $A^{-1}$.
3. Multiply $A^{-1}B$.
4. The solution is obtained from $X$.

Examples

1. Solve $$ \begin{cases} 2x+3y=5\\ x+4y=6 \end{cases} $$ using matrices.

   $$ A= \begin{pmatrix} 2&3\\ 1&4 \end{pmatrix}, \quad X= \begin{pmatrix} x\\y \end{pmatrix}, \quad B= \begin{pmatrix} 5\\6 \end{pmatrix} $$

   $$ A^{-1} = \frac15 \begin{pmatrix} 4&-3\\ -1&2 \end{pmatrix} $$

   $$ X=A^{-1}B $$

   $$ = \begin{pmatrix} \frac45&-\frac35\\ -\frac15&\frac25 \end{pmatrix} \begin{pmatrix} 5\\6 \end{pmatrix} $$

   $$ = \begin{pmatrix} \frac45(5)-\frac35(6)\\ -\frac15(5)+\frac25(6) \end{pmatrix} $$

   $$ = \begin{pmatrix} \frac25\\ \frac75 \end{pmatrix} $$

   Solution: $x=\frac25,\quad y=\frac75$

2. Solve $$ \begin{cases} 3x-y=8\\ -2x+y=-5 \end{cases} $$

   $$ A= \begin{pmatrix} 3&-1\\ -2&1 \end{pmatrix} $$

   $$ \det(A)=3(1)-(-1)(-2)=1 $$

   $$ A^{-1} = \begin{pmatrix} 1&1\\ 2&3 \end{pmatrix} $$

   $$ X= \begin{pmatrix} 1&1\\ 2&3 \end{pmatrix} \begin{pmatrix} 8\\-5 \end{pmatrix} $$

   $$ = \begin{pmatrix} 3\\1 \end{pmatrix} $$

   Solution: $x=3,\quad y=1$

3. Solve $$ \begin{cases} 4x+2y=10\\ 2x+3y=9 \end{cases} $$

   $$ A= \begin{pmatrix} 4&2\\ 2&3 \end{pmatrix} $$

   $$ \det(A)=4(3)-2(2)=8 $$

   $$ A^{-1} = \frac18 \begin{pmatrix} 3&-2\\ -2&4 \end{pmatrix} $$

   $$ X= \frac18 \begin{pmatrix} 3&-2\\ -2&4 \end{pmatrix} \begin{pmatrix} 10\\9 \end{pmatrix} $$

   $$ = \frac18 \begin{pmatrix} 12\\16 \end{pmatrix} = \begin{pmatrix} \frac32\\2 \end{pmatrix} $$

   Solution: $x=\frac32,\quad y=2$

Inconsistent and Dependent Systems

• Inconsistent system (no solution): $\det(A)=0$ but equations contradict.
• Dependent system (infinite solutions): $\det(A)=0$ and equations are multiples.

Example: Inconsistent System

Solve: $$ \begin{cases} x+2y=3\\ 2x+4y=7 \end{cases} $$

$$ \det(A)=1(4)-2(2)=0 $$

The second equation gives:

$$ 2(x+2y)=7 $$

but

$$ 2(3)=7 $$

which is false.

No solution.

Example: Dependent System

Solve: $$ \begin{cases} x+2y=3\\ 2x+4y=6 \end{cases} $$

The second equation is twice the first equation.

Infinite solutions:

$$ x=3-2t,\qquad y=t $$

Solving $3 \times 3$ Systems (Extension)

For a $3\times3$ system, inverse methods can be used, but calculations are longer.

• Gaussian elimination
• Cramer's rule
• Adjugate matrix method

Example: $3\times3$ System

Solve: $$ \begin{cases} x+y+z=6\\ x-y+z=2\\ 2x+y-z=1 \end{cases} $$

Matrix form:

$$ A= \begin{pmatrix} 1&1&1\\ 1&-1&1\\ 2&1&-1 \end{pmatrix} $$

$$ B= \begin{pmatrix} 6\\2\\1 \end{pmatrix} $$

Using inverse:

$$ A^{-1} = \begin{pmatrix} 0&\frac12&\frac12\\ \frac34&-\frac34&0\\ \frac14&\frac14&-\frac12 \end{pmatrix} $$

$$ X=A^{-1}B $$

$$ = \begin{pmatrix} \frac32\\ 3\\ \frac32 \end{pmatrix} $$

Solution: $$ x=\frac32,\quad y=3,\quad z=\frac32 $$

Summary Table

Concept	Key Points
Matrix order	$m\times n$ ($m$ rows, $n$ columns)
Addition/Subtraction	Same order, element-wise
Scalar multiplication	Multiply every entry by scalar
Matrix multiplication	$AB$ exists if columns of $A$ = rows of $B$
Multiplication rule	$(AB)_{ij}=row_i(A)\cdot column_j(B)$
Determinant	$\det(A)=ad-bc$
Inverse	$A^{-1}=\frac1{\det(A)} \begin{pmatrix} d&-b\\ -c&a \end{pmatrix}$
Solving $AX=B$	$X=A^{-1}B$
Singular matrix	$\det(A)=0$, no inverse
Inconsistent system	No solution
Dependent system	Infinite solutions