Matrix Algebra, Types & Operations

A matrix of order $m\times n$ has $m$ rows and $n$ columns.

• Equality: $A=B$ iff same order and $a_{ij}=b_{ij}$ for all $i,j$.
• Addition / scalar: both are entrywise, so addition needs matching order and $kA$ scales every entry by $k$.
• Linear systems: from $pA+qB$ and $rA+sB$ solve for $A$ and $B$ exactly as you would for two numbers — combine to cancel one matrix.
• Trace: $\operatorname{tr}(A)=a_{11}+a_{22}+\cdots+a_{nn}$ is the sum of the main-diagonal entries (square matrices only).

The product $A_{m\times n}B_{n\times p}=(AB)_{m\times p}$ exists only when $A$'s column count equals $B$'s row count, with entry $(AB)_{ij}=\sum_k a_{ik}b_{kj}$.

• Column extraction: $A\,e_j$ is the $j$-th column of $A$ (where $e_j$ is the column with 1 in position $j$, 0 elsewhere).
• Sum of all entries: $[1\ \cdots\ 1]\,B\,[1\ \cdots\ 1]^{T}$ equals the total of every entry of $B$.
• Not commutative: $AB\neq BA$ in general, and one of them may not even be defined.

$(A^{T})_{ij}=a_{ji}$, so an $m\times n$ matrix transposes to $n\times m$. Rules:

• Self-inverse: $(A^{T})^{T}=A$
• Sum / scalar: $(A+B)^{T}=A^{T}+B^{T}$, $(kA)^{T}=kA^{T}$
• Reversal: $(AB)^{T}=B^{T}A^{T}$
• Symmetric / skew: $A^{T}=A$ (symmetric) or $A^{T}=-A$ (skew — its diagonal is all zeros).

To exploit a transpose relation, transpose the whole equation and substitute back.

Because multiplication does not commute:

• $(A+B)^{2}=A^{2}+AB+BA+B^{2}$ — the middle collapses to $2AB$ only when $AB=BA$.
• $(A+B)(A-B)=A^{2}-AB+BA-B^{2}$, which equals $A^{2}-B^{2}$ only when $A,B$ commute.
• Zero divisors: $AB=O$ is possible with $A\neq O$ and $B\neq O$. If both are nonzero $n\times n$ and $AB=O$, then $\det A=0$ and $\det B=0$ (both singular), so $AX=0$ has non-trivial (infinitely many) solutions.
• Commuting count: requiring $AB=BA$ for a fixed $A$ imposes linear constraints on $B$'s entries; the free entries then range over the allowed value set.

The three elementary row operations on a matrix are:

• Interchange: $R_i\leftrightarrow R_j$ (changes two rows).
• Scaling: $R_i\to kR_i$ with $k\neq0$ (changes one row).
• Row addition: $R_i\to R_i+kR_j$ (changes one row, using a multiple of another).

A matrix is obtainable in one operation iff at most the changed row(s) differ from the original by exactly one of these patterns. To test a 'row-addition' candidate, solve for the multiplier $k$ on one entry and check it is consistent on the rest of the row.

• Free-choice count: an $m\times n$ matrix with each entry from a $k$-element set gives $k^{mn}$ matrices.
• **Singular $2\times2$:** $\det=ad-bc=0\iff ad=bc$. Count ordered pairs by product: if $n_p$ pairs from the set have product $p$, the number of singular matrices is $\sum_p n_p^{2}$.
• Trace bridge: $\operatorname{tr}(A^{T}A)=\sum_{i,j}a_{ij}^{2}$. Over entries in $\{-1,0,1\}$ each square is $0$ or $1$, so the trace simply counts the nonzero entries — choose their positions, then their signs.