Matrices: addition, multiplication, transpose

Addition: same dimension

$A + B$ makes sense only if $A$ and $B$ have the same dimension. Add coefficient by coefficient.

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$

Properties: commutative, associative, neutral element $0$ (zero matrix).

Scalar multiplication

$\lambda A$: multiply every coefficient by $\lambda$.

$3 \times \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix}$

Matrix product: compatible dimensions

$A \times B$ makes sense iff columns of $A$ = rows of $B$.

If $A$ is $(n, p)$ and $B$ is $(p, q)$, then $AB$ is $(n, q)$.

Formula: $(AB)_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$ (row $i$ of $A$ × column $j$ of $B$).

Example $(2,2) \times (2,2)$

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}$

Detail: $1 \times 5 + 2 \times 7 = 19$, $1 \times 6 + 2 \times 8 = 22$, etc.

⚠️ Non-commutativity

$AB \neq BA$ in general! Even when both products exist.

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

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

Transpose: $A^T$

Swap rows and columns. If $A$ is $(n, p)$, $A^T$ is $(p, n)$.

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

Properties:

• $(A + B)^T = A^T + B^T$
• $(\lambda A)^T = \lambda A^T$
• $(AB)^T = B^T A^T$ (reversed order!)
• $(A^T)^T = A$

Common mistakes

1. Adding $A(2,3) + B(3,2)$: different dimensions, impossible.
2. Writing $AB = BA$: false except in special cases.
3. $(AB)^T = A^T B^T$: NO, it’s $B^T A^T$.
4. Forgetting row × column in the matrix product.

?À toi de jouer

A is (3, 2) and B is (2, 4). Dimension of AB?

A.(3, 4)B.(2, 2)C.(4, 3)D.Impossible

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