Matrix and Determinant

1. Matrix

1.1 What is a Matrix?

A matrix is a rectangular arrangement of numbers (or functions) enclosed in brackets.

• Order (Dimension): If a matrix has $m$ rows and $n$ columns, its order is $m \times n$ (read as "m by n").
• General Representation:
  $A = [a_{ij}]_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}$
  
  Here, $a_{ij}$ represents the element in the $i$-th row and $j$-th column.

1.2 Some Special Types of Matrices

1. Row Matrix: Has only 1 row. (e.g., $[1, 2, 3]_{1 \times 3}$)
2. Column Matrix: Has only 1 column. (e.g., $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}_{3 \times 1}$)
3. Square Matrix: Number of rows = Number of columns (Order $n \times n$). Only square matrices have determinants.
4. Diagonal Matrix: A square matrix where all non-diagonal elements are zero. (e.g., $\begin{bmatrix} 2 & 0 \\ 0 & 5 \end{bmatrix}$).
5. Scalar Matrix: A diagonal matrix where all diagonal elements are equal. (e.g., $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$).
6. Identity Matrix ($I$): A scalar matrix with diagonal elements exactly $1$. Acts like the number '1' in multiplication.
   $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
7. Zero Matrix ($O$): All elements are zero.
8. Triangular Matrix:
   • Upper Triangular: All elements below the main diagonal are zero.
   • Lower Triangular: All elements above the main diagonal are zero.

1.3 Transpose of a Matrix

The transpose of $A$, denoted by $A^T$, is obtained by interchanging its rows and columns.

• If A = [a_{ij}]_{m \times n}, \text{ then } A^T = [a_{ji}]_{n \times m}

Example:
If $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2 \times 3} \text{ then } A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}_{3 \times 2}$.

Symmetric vs. Skew-Symmetric:

• Symmetric: $A^T = A$. (e.g., $\begin{bmatrix} 1 & x \ x & 2 \end{bmatrix}$).
• Skew-Symmetric: $A^T = -A$. (Diagonal elements must be zero).

1.4 Properties of Transpose Matrix

Let $A$ and $B$ be matrices of appropriate orders, and $k$ be a scalar.

1. $(A^T)^T = A$
2. $(A + B)^T = A^T + B^T$
3. $(kA)^T = kA^T$
4. $(AB)^T = B^T A^T$ (Reversal Law – order reverses!)

2. Determinant

2.1 What is a Determinant?

A determinant is a scalar value computed from a square matrix, denoted by $|A|$ or $\det(A)$.

For a $2 \times 2$ Matrix:
If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $|A| = ad - bc$.

For a $3 \times 3$ Matrix: We expand using minors and cofactors.

2.2 Minors and Cofactors

• Minor ($M_{ij}$): Determinant of the sub-matrix formed by deleting the $i$-th row and $j$-th column.
• Cofactor ($C_{ij}$): The signed minor.
  $C_{ij} = (-1)^{i+j} M_{ij}$

Sign Rule for Cofactors:
$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$

2.3 Sarrus Rule (Only for $3 \times 3$ Matrices)

Let $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$.

Method: Repeat the first two columns to the right, then compute:
$|A| = (aei + bfg + cdh) - (ceg + afh + bdi)$

2.4 Properties of Determinants (Time-Savers)

1. Reflection: $|A| = |A^T|$
2. Row/Column Interchange: If two rows (or columns) are interchanged, the determinant changes sign: $|A_{new}| = -|A|$.
3. Identical Rows/Columns: If two rows (or columns) are identical, $|A| = 0$.
4. Scalar Multiplication: If all elements of a row (or column) are multiplied by $k$, the determinant becomes $k$ times the original: $|A_{new}| = k|A|$.
5. Sum Property: If elements of a row are sums, the determinant splits into sum of determinants.
6. Invariance Property: Adding a multiple of one row to another row (or column to column) does not change the determinant. (Most powerful!)

3. Adjoint and Inverse

3.1 Adjoint of a Matrix

The adjoint of a square matrix $A$, denoted by $adj(A)$, is the transpose of the cofactor matrix.

Steps:

1. Find the cofactor $C_{ij}$ for every element.
2. Arrange them in a matrix (Cofactor Matrix).
3. Take its transpose: $adj(A) = (Cofactor , Matrix)^T$.

Fundamental Identity:
$A \cdot adj(A) = adj(A) \cdot A = |A| \cdot I$
(where $I$ is the Identity Matrix).

3.2 Inverse of a Matrix

If A and B are square matrices such that $AB = BA = I$, then A is called the inverse matrix of B. The inverse of $A$, denoted by $A^{-1}$, exists only if $A$ is non-singular ($|A| \neq 0$).

The Golden Formula:
$A^{-1} = \frac{1}{|A|} \cdot adj(A)$

Conditions for Existence:

• Matrix must be square.
• $|A| \neq 0$.

Verification:
$A \cdot A^{-1} = A^{-1} \cdot A = I$

Summary Table for Quick Revision