Unit 2 - Notes

MTH005

Unit 2: Matrices

1. Introduction to Matrices

1.1 Definition

A matrix is an ordered rectangular array of numbers (real or complex) or functions. These numbers or functions are called the elements or the entries of the matrix.

A matrix is denoted by capital letters ( $A$ , $B$ , $X$ , etc.) and enclosed by square brackets $[ ]$ or parentheses $( )$ .

General Form:

A = \begin{bmatrix} 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{bmatrix}

1.2 Order of a Matrix

A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$ (read as "m by n").

Rows: Horizontal lines of elements.
Columns: Vertical lines of elements.

1.3 Types of Matrices

Row Matrix: Has only one row ( $1 \times n$ ).
Column Matrix: Has only one column ( $m \times 1$ ).
Square Matrix: Number of rows equals number of columns ( $m = n$ ).
Diagonal Matrix: A square matrix where all non-diagonal elements are zero ( $a_{ij} = 0$ for $i \neq j$ ).
Scalar Matrix: A diagonal matrix where all diagonal elements are equal.
Identity (Unit) Matrix ( $I$ ): A square matrix where diagonal elements are $1$ and others are $0$.
Null (Zero) Matrix ( $O$ ): All elements are zero.

2. Operations on Matrices

2.1 Addition and Subtraction

Two matrices $A$ and $B$ can be added or subtracted only if they are of the same order.

Addition ( $A + B$ ): Add corresponding elements.
$[a_{ij}] + [b_{ij}] = [a_{ij} + b_{ij}]$
Subtraction ( $A - B$ ): Subtract corresponding elements.
$[a_{ij}] - [b_{ij}] = [a_{ij} - b_{ij}]$

Properties:

Commutative: $A + B = B + A$
Associative: $(A + B) + C = A + (B + C)$

2.2 Scalar Multiplication

If a matrix $A$ is multiplied by a scalar constant $k$ , then every element of $A$ is multiplied by $k$ .

kA = [k \cdot a_{ij}]

2.3 Multiplication of Matrices

The product $AB$ is defined only if the number of columns in $A$ equals the number of rows in $B$ .

If $A$ is $m \times n$ and $B$ is $n \times p$ , the product $AB$ is of order $m \times p$ .

Process (Row-by-Column Multiplication):
The element in the $i$ -th row and $j$ -th column of the product is the sum of the products of the corresponding elements of the $i$ -th row of $A$ and the $j$ -th column of $B$ .

(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}

Example ( $2 \times 2$ ):

\begin{bmatrix} a & b \\ c & d \end{bmatrix} \times \begin{bmatrix} w & x \\ y & z \end{bmatrix} = \begin{bmatrix} aw+by & ax+bz \\ cw+dy & cx+dz \end{bmatrix}

Important Properties:

Not Commutative: Generally, $AB \neq BA$ .
Associative: $(AB)C = A(BC)$ .
Distributive: $A(B + C) = AB + AC$ .

3. Transpose of a Matrix

3.1 Definition

The transpose of a matrix $A$ , denoted by $A^T$ or $A'$ , is obtained by interchanging the rows and columns of $A$ .
If $A = [a_{ij}]_{m \times n}$ , then $A^T = [a_{ji}]_{n \times m}$ .

Example:
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ , then $A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$ .

3.2 Properties of Transpose

$(A^T)^T = A$
$(kA)^T = kA^T$ (where $k$ is a scalar)
$(A + B)^T = A^T + B^T$
Reversal Law: $(AB)^T = B^T A^T$

3.3 Symmetric and Skew-Symmetric Matrices

Symmetric: If $A^T = A$ (requires $a_{ij} = a_{ji}$ ).
Skew-Symmetric: If $A^T = -A$ (requires $a_{ij} = -a_{ji}$ and diagonal elements must be 0).

4. Determinants

4.1 Definition

To every square matrix $A$ of order $n$ , we can associate a number (real or complex) called the determinant of the matrix $A$ , denoted by $\det(A)$ , $|A|$ , or $\Delta$ .

4.2 Evaluation of Determinants

Order 1:
If $A = [a]$ , then $|A| = a$ .

Order 2:

|A| = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = (a_{11}a_{22}) - (a_{21}a_{12})

Cross-multiply: (Main diagonal product) - (Off-diagonal product).

Order 3:

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}

We expand along the first row (applying signs

+

-

+

|A| = a_{11}\begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13}\begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

|A| = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22})

4.3 Singular and Non-Singular Matrices

Singular Matrix: $|A| = 0$ (Inverse does not exist).
Non-Singular Matrix: $|A| \neq 0$ (Inverse exists).

5. Properties of Determinants

These properties simplify the evaluation of higher-order determinants.

Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged. $|A| = |A^T|$ .
Switching Property: If any two rows (or columns) are interchanged, the sign of the determinant changes.
Repetition Property: If any two rows (or columns) are identical, the value of the determinant is zero.
Scalar Property: If each element of a row (or column) is multiplied by a constant , then the value of the new determinant is times the original determinant.
- Note: $|kA| = k^n |A|$ for an $n \times n$ matrix.
Summation Property: If some or all elements of a row (or column) are expressed as the sum of two (or more) terms, the determinant can be expressed as the sum of two (or more) determinants.
Invariance Property: The value of a determinant remains unchanged if we add to the elements of any row (or column) the equimultiples of corresponding elements of other rows (or columns).
- Operation: $R_i \to R_i + kR_j$
Triangle Property: If all elements below or above the main diagonal are zero, the determinant is the product of the diagonal elements.

6. Inverse of a Matrix (Adjoint Method)

The inverse of a square matrix $A$ is a matrix $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ .

6.1 Steps to Find Inverse

To find $A^{-1}$ for a $3 \times 3$ matrix:

Check Determinant: Calculate $|A|$ . If $|A| = 0$ , inverse does not exist.
Find Minors: Find the minor ( $M_{ij}$ ) for every element. The minor of $a_{ij}$ is the determinant of the submatrix left after deleting the $i$ -th row and $j$ -th column.
Find Cofactors: Calculate cofactors ( $C_{ij}$ or $A_{ij}$ ) using the formula:
$C_{ij} = (-1)^{i+j} M_{ij}$
(This applies a checkerboard sign pattern: $+ - +$ / $- + -$ / $+ - +$ ).
Matrix of Cofactors: Arrange the cofactors into a matrix $C$ .
Adjoint Matrix ( $Adj \ A$ ): Take the transpose of the cofactor matrix.
$Adj \ A = [C_{ij}]^T$
Apply Formula:
$A^{-1} = \frac{1}{|A|} (Adj \ A)$

7. Worked Example: Inverse of a 3x3 Matrix

Problem: Find the inverse of $A = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 2 & 3 \\ 1 & 0 & 1 \end{bmatrix}$ .

Step 1: Calculate Determinant ( $|A|$ )

Expanding along Row 1:

|A| = 1(2(1) - 3(0)) - 2(0(1) - 3(1)) + 1(0(0) - 2(1))

|A| = 1(2) - 2(-3) + 1(-2)

|A| = 2 + 6 - 2 = 6

Since

6 \neq 0

, the inverse exists.

Step 2 & 3: Find Cofactors ( $C_{ij}$ )

Sign pattern: $\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$

$C_{11} = +(2(1) - 3(0)) = 2$
$C_{12} = -(0(1) - 3(1)) = -(-3) = 3$
$C_{13} = +(0(0) - 2(1)) = -2$
$C_{21} = -(2(1) - 1(0)) = -2$
$C_{22} = +(1(1) - 1(1)) = 0$
$C_{23} = -(1(0) - 2(1)) = -(-2) = 2$
$C_{31} = +(2(3) - 1(2)) = 6 - 2 = 4$
$C_{32} = -(1(3) - 1(0)) = -3$
$C_{33} = +(1(2) - 2(0)) = 2$

Step 4: Matrix of Cofactors

C = \begin{bmatrix} 2 & 3 & -2 \\ -2 & 0 & 2 \\ 4 & -3 & 2 \end{bmatrix}

Step 5: Adjoint Matrix ( $Adj \ A$ )

Transpose the Cofactor Matrix:

Adj \ A = C^T = \begin{bmatrix} 2 & -2 & 4 \\ 3 & 0 & -3 \\ -2 & 2 & 2 \end{bmatrix}

Step 6: Inverse Matrix ( $A^{-1}$ )

A^{-1} = \frac{1}{|A|} (Adj \ A) = \frac{1}{6} \begin{bmatrix} 2 & -2 & 4 \\ 3 & 0 & -3 \\ -2 & 2 & 2 \end{bmatrix}

A^{-1} = \begin{bmatrix} \frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \\ \frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{bmatrix}