A matrix is a rectangular array of numbers arranged in rows and columns

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

• Order/Dimension: \( m \times n \) (m rows by n columns)

• Element: \( a_{ij} \) is the element in row i, column j

• Square Matrix: Same number of rows and columns (n × n)

• Row Matrix: Matrix with only one row (1 × n)

• Column Matrix: Matrix with only one column (m × 1)

• Zero Matrix: All elements are zero

• Identity Matrix (I): Square matrix with 1's on diagonal, 0's elsewhere

• Addition/Subtraction: Matrices must have the SAME dimensions

• Scalar Multiplication: Can multiply any matrix by a scalar

• Matrix Multiplication: Number of columns in first = number of rows in second

Add or subtract corresponding elements (element-by-element)

\[ A \pm B = \begin{bmatrix} a_{11} \pm b_{11} & a_{12} \pm b_{12} \\ a_{21} \pm b_{21} & a_{22} \pm b_{22} \end{bmatrix} \]

⚠️ Both matrices MUST have the same dimensions

Calculate: \( \begin{bmatrix} 3 & 5 \\ 1 & -2 \end{bmatrix} + \begin{bmatrix} 2 & -1 \\ 4 & 6 \end{bmatrix} \)

Multiply every element by the scalar

\[ kA = k\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} ka_{11} & ka_{12} \\ ka_{21} & ka_{22} \end{bmatrix} \]

Calculate: \( 3\begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} \)

For \( A_{m \times n} \cdot B_{p \times q} \):

• Must have: \( n = p \) (columns of A = rows of B)

• Result will be: \( (AB)_{m \times q} \)

Element \( c_{ij} \) in result = (row i of A) · (column j of B)

\[ c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj} = \sum_{k=1}^{n} a_{ik}b_{kj} \]

Calculate: \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)

A scalar value calculated from a square matrix. Notation: det(A) or |A|

For 2×2 Matrix:

\[ \det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc \]

For 3×3 Matrix:

\[ \det\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Or: \( aei + bfg + cdh - ceg - bdi - afh \)

Find: \( \det\begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix} \)

For square matrix A, the inverse \( A^{-1} \) satisfies:

\[ AA^{-1} = A^{-1}A = I \]

⚠️ Matrix must be square and det(A) ≠ 0

A matrix is invertible if and only if:

✓ It is a square matrix

✓ Its determinant ≠ 0

\[ A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \]

where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) and \( ad - bc \neq 0 \)

Find inverse: \( A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix} \)

\[ A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A) \]

where adj(A) is the adjugate (transpose of cofactor matrix)

This method involves finding minors and cofactors

For equation \( AX = B \):

\[ X = A^{-1}B \]

Steps:

1. Find \( A^{-1} \)

2. Multiply: \( X = A^{-1}B \)

For \( X + A = B \): Subtract A from both sides

\[ X = B - A \]

Solve: \( 2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \)

To transform a figure, organize vertices as columns:

\[ V = \begin{bmatrix} x_1 & x_2 & x_3 & \cdots \\ y_1 & y_2 & y_3 & \cdots \end{bmatrix} \]

Transformed vertices: \( V' = T \cdot V \) (where T is transformation matrix)

Reflect triangle with vertices A(1,2), B(3,4), C(5,1) over x-axis