Matrix Operations

Chapter 2 • Section 2-1

"Why was the matrix always tired? Because it had too many rows! 😴"

🔢 Definitions

Matrix Size ($m \times n$)

A rectangular array with $m$ rows and $n$ columns.

$$ A = \begin{bmatrix} a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \dots & a_{mn} \end{bmatrix} $$

Row Matrix

Size $1 \times n$. Example: $\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$

Column Matrix

Size $m \times 1$. Example: $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$

Square Matrix

Size $n \times n$ (Same number of rows and columns).

Basic Operations

1 Matrix Addition

Add corresponding entries. Must be same size!

$$ \begin{bmatrix} 1 & 3 \\ 2 & 5 \end{bmatrix} + \begin{bmatrix} 0 & -2 \\ 6 & 1 \end{bmatrix} = \begin{bmatrix} 1+0 & 3+(-2) \\ 2+6 & 5+1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 8 & 6 \end{bmatrix} $$

2 Scalar Multiplication

Multiply every entry by the scalar $k$.

$$ 3 \begin{bmatrix} 2 & 0 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 3(2) & 3(0) \\ 3(3) & 3(1) \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 9 & 3 \end{bmatrix} $$

Properties

  • $k(A + B) = kA + kB$ (Distributive)
  • $(k + p)A = kA + pA$ (Distributive)
  • $k(pA) = (kp)A$ (Associative)
  • $1A = A$ (Identity)

Example: Simplify

Simplify $2(A - 3B) + 3(2B + A)$.

$= 2A - 6B + 6B + 3A$
$= 5A$

🧮 Quick Calc Win $10

Calculate: $2\begin{bmatrix} 1 \\ 2 \end{bmatrix} - \begin{bmatrix} 2 \\ 4 \end{bmatrix}$

The Transpose ($A^T$)

Swap rows and columns! The $(i, j)$ entry becomes the $(j, i)$ entry.

Original Matrix $A$

$$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} $$

Size: $2 \times 3$

Transpose $A^T$

$$ \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} $$

Size: $3 \times 2$

Properties

  • $(A^T)^T = A$ (Flipping twice gets you back)
  • $(A+B)^T = A^T + B^T$ (Transpose distributes over addition)
  • $(kA)^T = kA^T$ (Scalars don't care about transpose)
  • $(AB)^T = B^T A^T$ (The "Shoe-Sock" Rule: Reverse the order!)

Symmetric & Skew-Symmetric

Symmetric Matrix

$A^T = A$

Mirror image across the main diagonal.

$$ \begin{bmatrix} 1 & \color{blue}2 & \color{red}3 \\ \color{blue}2 & 4 & \color{green}5 \\ \color{red}3 & \color{green}5 & 6 \end{bmatrix} $$

Skew-Symmetric Matrix

$A^T = -A$

Diagonal must be 0. Off-diagonals are negatives.

$$ \begin{bmatrix} 0 & \color{blue}2 \\ \color{blue}-2 & 0 \end{bmatrix} $$

🏆 Master Challenge Win $20

If $A$ is a square matrix, is $A - A^T$ always skew-symmetric?