Has a unique solution if and only if $r=n$.

Can be solved by Gauss-Jordan elimination to directly obtain the solution vector $\vec{x}$, where $[I|\vec{x}]$ is the reduced augmented matrix.

Matrices

Definition (Matrix)

A rectangular array of $m \times n$ numbers, $A = \begin{pmatrix} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \dots & a_{mn} \end{pmatrix}$.

$m$: number of rows. $n$: number of columns.
$a_{ij}$: entry in row $i$, column $j$.
$M_{m \times n}$: set of all matrices of size $m \times n$.
A vector is a matrix with one column (column vector) or one row (row vector).

Equality of Matrices

Two matrices $A$ and $B$ are equal if they have the same size and $a_{ij} = b_{ij}$ for all $i, j$.

Matrix Operations: Addition and Subtraction

For $A = (a_{ij})$ and $B = (b_{ij})$ of the same size $m \times n$:

$A \pm B = (a_{ij} \pm b_{ij})$ (element-wise operation).

Matrix Operations: Multiplication with a Scalar

For $A = (a_{ij})$ of size $m \times n$ and scalar $\lambda \in \mathbb{R}$:

$\lambda \cdot A = (\lambda a_{ij})$ (element-wise multiplication).

Properties (Calculation Rules)

For $m \times n$ matrices $A, B, C$ and scalars $\lambda, \mu \in \mathbb{R}$:

$A + B = B + A$ (Commutative)
$(A + B) + C = A + (B + C)$ (Associative)
$\lambda (A + B) = \lambda A + \lambda B$ (Distributive)
$(\lambda + \mu) A = \lambda A + \mu A$ (Distributive)

Matrix Operations: Matrix Multiplication

For $A$ of size $m \times n$ and $B$ of size $n \times p$, their product $C = AB$ is an $m \times p$ matrix with entries $c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$.

$AB$ is defined only if the number of columns of $A$ equals the number of rows of $B$.
The product of an $m \times 1$ column vector and a $1 \times m$ row vector is an $m \times m$ matrix.
The product of a $1 \times m$ row vector and an $m \times 1$ column vector is a $1 \times 1$ matrix (scalar, dot product).

Missing Calculation Rules for Matrix Multiplication

Not commutative: $AB \neq BA$ in general.
No zero law for products: $AB = 0$ does not imply $A=0$ or $B=0$.
No cancellation law: $AC = BC$ and $C \neq 0$ does not imply $A=B$.

Matrix Operations: Transposing Matrices

The transpose of an $m \times n$ matrix $A$, denoted $A^T$, is an $n \times m$ matrix where the rows of $A$ become the columns of $A^T$ (i.e., $(A^T)_{ij} = A_{ji}$).

Properties of Transposes

For matrices $A, B$ and scalar $c$:

$(A^T)^T = A$
$(A + B)^T = A^T + B^T$
$(cA)^T = cA^T$
$(AB)^T = B^T A^T$

Some Special Matrices

Zero Matrix ($0$): All entries are 0. ($a_{ij}=0$ for all $i, j$).
Properties of Zero Matrices:
- $A + 0_{m \times n} = A$
- $A + (-A) = 0_{m \times n}$
- If $cA = 0_{m \times n}$, then $c=0$ or $A=0_{m \times n}$.
Square Matrix: An $n \times n$ matrix. Main diagonal entries are $a_{ii}$.
Diagonal Matrix: A square matrix where $d_{ij} = 0$ for $i \neq j$. $D = \begin{pmatrix} d_{11} & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & d_{nn} \end{pmatrix}$.
Identity Matrix ($I$): A diagonal matrix with main diagonal entries equal to 1. ($d_{ii}=1$). Denoted $I_{n \times n}$ or $I$.
Upper Triangular Matrix: A square matrix where $d_{ij} = 0$ for $i > j$.
Lower Triangular Matrix: A square matrix where $d_{