Introduction to Matrices

• A matrix is a rectangular array of numbers or symbols.
• An $m \times n$ matrix has $m$ rows and $n$ columns.

Matrix Examples

• $\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix.
• $\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}$ is a $3 \times 1$ matrix.
• $\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$, $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, $\begin{bmatrix} 1 \ 0 \ 1 \end{bmatrix}$, $\begin{bmatrix} 1 & 2 \end{bmatrix}$ are valid matrices

Special Matrices

• Square Matrix: Same number of rows and columns, example $\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$
• Identity Matrix: Denoted by $I$, a square matrix with 1s on the main diagonal and 0s elsewhere, example $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$
• Zero Matrix: All entries are 0 , example $\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}$
• Diagonal Matrix: Square matrix in which entries outside the main diagonal are all zero, example: $\begin{bmatrix} 1 & 0 \ 0 & 2 \end{bmatrix}$
• Upper Triangular Matrix: Matrix in which all entries below the main diagonal are zero, example: $\begin{bmatrix} 1 & 2 \ 0 & 3 \end{bmatrix}$
• Lower Triangular Matrix: Matrix in which all entries above the main diagonal are zero, example: $\begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix}$
• Transpose: Given a matrix A, its transpose, denoted by $A^T$, is obtained by interchanging rows and columns.
  • If $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, then $A^T = \begin{bmatrix} 1 & 3 \ 2 & 4 \end{bmatrix}$

Matrix Operations

• Matrices can be added or subtracted if they have the same dimensions.
• If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$ matrices, then $A + B = [a_{ij} + b_{ij}]$ and $A - B = [a_{ij} - b_{ij}]$.
• If $A = [a_{ij}]$ is a matrix and $c$ is a scalar, then $cA = [ca_{ij}]$.
• If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, then the product $AB$ is an $m \times p$ matrix.
• The $(i, j)$-th entry of $AB$ is given by: $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$

Properties of Matrix Multiplication

• Associative: $(AB)C = A(BC)$
• Distributive: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$
• Identity: $AI = IA = A$

Determinant

• The determinant is a scalar value that can be computed from the elements of a square matrix, encodes certain properties of the linear transformation described by the matrix.
• The determinant of a matrix $A$ is denoted as det$(A)$ or $|A|$.

$2 \times 2$ Matrix

• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is: $det(A) = ad - bc$

$3 \times 3$ Matrix

• For a $3 \times 3$ matrix, computed using the rule of Sarrus or cofactor expansion.
• Example using cofactor expansion: $A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$
• $det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

Properties of Determinants

• $det(I) = 1$
• $det(A^T) = det(A)$
• If $A$ has a row or column of zeros, then $det(A) = 0$
• If $A$ has two identical rows or columns, then $det(A) = 0$
• If $B$ is obtained from $A$ by interchanging two rows or columns, then $det(B) = -det(A)$
• If $B$ is obtained from $A$ by multiplying a row or column by a scalar $k$, then $det(B) = k \cdot det(A)$
• $det(AB) = det(A) \cdot det(B)$

Inverse

• The inverse of a square matrix $A$, denoted as $A^{-1}$, is a matrix such that $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

$2 \times 2$ Matrix

• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the inverse is: $A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$, only if $ad - bc \neq 0$

Properties of Inverses

• $(A^{-1})^{-1} = A$
• $(cA)^{-1} = \frac{1}{c}A^{-1}$
• $(AB)^{-1} = B^{-1}A^{-1}$
• $(A^T)^{-1} = (A^{-1})^T$
• A matrix is invertible if and only if $det(A) \neq 0$.