3: Matrix operations

Section 1: Systems and Matrices - 3: Matrix Operations

Matrix operations extend arithmetic to arrays of numbers, forming the computational backbone of linear algebra. Understanding these rules is essential for solving linear systems and beyond.

Addition and Subtraction: Matrices must be the same size ($m$ rows $\times$ $n$ columns). Add or subtract corresponding elements. For matrices $\mathbf{A}$ and $\mathbf{B}$:

$$\mathbf{A} + \mathbf{B} = [a_{ij} + b_{ij}]$$ $$\mathbf{A} - \mathbf{B} = [a_{ij} - b_{ij}]$$

Properties include commutativity ($\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$) and associativity ($\mathbf{A} + (\mathbf{B} + \mathbf{C}) = (\mathbf{A} + \mathbf{B}) + \mathbf{C}$).

Scalar Multiplication: Multiply every element of a matrix by a scalar (real number) $c$:

$$c\mathbf{A} = [c \cdot a_{ij}]$$

Scalars distribute over matrix addition: $c(\mathbf{A} + \mathbf{B}) = c\mathbf{A} + c\mathbf{B}$.

Matrix Multiplication: The product $\mathbf{AB}$ is defined only if $\mathbf{A}$ is $m \times n$ and $\mathbf{B}$ is $n \times p$ (columns of $\mathbf{A}$ = rows of $\mathbf{B}$). The result is an $m \times p$ matrix. The element in row $i$, column $j$ of $\mathbf{AB}$ is computed by taking the dot product of row $i$ of $\mathbf{A}$ and column $j$ of $\mathbf{B}$:

$$(\mathbf{AB})_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} \quad \text{(sum from } k=1 \text{ to } n\text{)}$$

Key properties:

• Associative: $(\mathbf{AB})\mathbf{C} = \mathbf{A}(\mathbf{BC})$
• Distributive: $\mathbf{A}(\mathbf{B}+\mathbf{C}) = \mathbf{AB} + \mathbf{AC}$ and $(\mathbf{A}+\mathbf{B})\mathbf{C} = \mathbf{AC} + \mathbf{BC}$
• NOT Commutative: $\mathbf{AB} \neq \mathbf{BA}$ in general (even if both products exist).
• Identity Matrix: The $n \times n$ identity matrix $\mathbf{I}_n$ (1s on diagonal, 0s elsewhere) acts like 1: $\mathbf{A}\mathbf{I} = \mathbf{A}$ and $\mathbf{I}\mathbf{A} = \mathbf{A}$ for compatible $\mathbf{A}$.

Transpose: The transpose of an $m \times n$ matrix $\mathbf{A}$, denoted $\mathbf{A}^\top$, is the $n \times m$ matrix formed by swapping rows and columns: $(\mathbf{A}^\top)_{ij} = a_{ji}$. Properties include:

$$(\mathbf{A}^\top)^\top = \mathbf{A}$$ $$(\mathbf{A} + \mathbf{B})^\top = \mathbf{A}^\top + \mathbf{B}^\top$$ $$(c\mathbf{A})^\top = c \mathbf{A}^\top$$ $$(\mathbf{AB})^\top = \mathbf{B}^\top \mathbf{A}^\top \quad \text{(note the order reversal!)}$$

Critical Considerations:

• Dimension Mismatch: Attempting addition/subtraction on different-sized matrices or multiplication where columns of first $\neq$ rows of second is undefined.
• Zero Matrix: The matrix $\mathbf{0}$ (all zeros) acts as the additive identity: $\mathbf{A} + \mathbf{0} = \mathbf{A}$.