Matrix Operations

By the end of this lesson, you will review and extend matrix operations, including multiplication of larger matrices, and understand matrix algebra as a system distinct from real-number algebra.

Quick Review of Matrix Operations

Addition/Subtraction: requires same dimensions; add corresponding elements.

Scalar multiplication: multiply every entry by the scalar.

Matrix multiplication: $C_{ij} = \sum_k A_{ik}B_{kj}$ (dot product of row $i$ and column $j$).

Dimensions: $(m \times n)(n \times p) = (m \times p)$; inner dimensions must match.

Example 1 — $3 \times 3$ Multiplication

$$\begin{pmatrix}1&2&0\3&1&4\2&0&5\end{pmatrix}\begin{pmatrix}2\1\3\end{pmatrix} = \begin{pmatrix}4\13\19\end{pmatrix}$$

Row 1: $1(2)+2(1)+0(3) = 4$; Row 2: $3(2)+1(1)+4(3) = 19$... let me recompute:

Row 2: $3(2) + 1(1) + 4(3) = 6 + 1 + 12 = 19$ ✓

Row 3: $2(2) + 0(1) + 5(3) = 4 + 0 + 15 = 19$ — but Row 2 is also 19... let me verify Row 2: $6 + 1 + 12 = 19$ ✓ and Row 3: $4 + 15 = 19$ ✓ (coincidence).

Transpose

The transpose $A^T$ flips rows and columns: $(A^T){ij} = A{ji}$.

$$A = \begin{pmatrix}1&2&3\4&5&6\end{pmatrix} \implies A^T = \begin{pmatrix}1&4\2&5\3&6\end{pmatrix}$$

Symmetric matrix: $A = A^T$ (square matrices where entry $(i,j) = $ entry $(j,i)$).

Powers of a Matrix

$A^2 = A \cdot A$, $A^3 = A \cdot A^2$, etc. (only defined for square matrices).

The identity matrix $I$ satisfies $AI = IA = A$ and $A^0 = I$.

Non-Commutativity and Other Differences from Real Numbers

In real arithmetic: $ab = ba$ (commutative) and $ab = 0 \implies a = 0$ or $b = 0$.

For matrices:

• $AB \neq BA$ in general
• $AB = O$ (zero matrix) does NOT require $A = O$ or $B = O$

Counterexample to zero-product:

$\begin{pmatrix}1&1\1&1\end{pmatrix}\begin{pmatrix}1&-1\-1&1\end{pmatrix} = \begin{pmatrix}0&0\0&0\end{pmatrix}$

Neither factor is zero, yet the product is.

Summary

Matrix multiplication: $(m \times n)(n \times p) = (m \times p)$; row × column dot products.

Transpose: $A^T$ — flip rows and columns.

Key differences from scalars: not commutative; zero product doesn't imply zero factor.

Identity matrix: $I_n$ — the multiplicative identity for $n\times n$ matrices.