Systems, Matrices, and Determinants

A solution of a system is an ordered pair (or triple) that satisfies every equation at once. Two main hand methods:

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• Substitution: solve one equation for a variable, then plug into the other. Best when a variable already has coefficient $1$, or for nonlinear systems.
• Elimination: add multiples of the equations to cancel a variable. Best for linear systems with awkward coefficients.

A linear system may have one solution (lines cross), no solution (parallel lines; you reach a false statement like $0=5$), or infinitely many (same line; you reach a true statement like $0=0$). A nonlinear system (line and circle, two conics, …) can have $0,1,2,$ or more solutions.

Write the system as an augmented matrix and use elementary row operations to reach row-echelon form (leading $1$'s stepping to the right, zeros below), then back-substitute.

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• Swap two rows: $R_i \leftrightarrow R_j$.
• Scale a row: $R_i \to kR_i$ ($k\neq 0$).
• Add a multiple of one row to another: $R_i \to R_i + kR_j$.

Reduced row-echelon form additionally clears entries above each leading $1$ (Gauss--Jordan).

For matrices of the same size, add/subtract entrywise; scalar multiply multiplies every entry.

Matrix product $AB$ is defined only when $A$ is $m\times n$ and $B$ is $n\times p$ (inner dimensions match); the result is $m\times p$. Entry $(i,j)$ of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$. In general $AB\neq BA$.

$A^{-1}$ satisfies $AA^{-1}=A^{-1}A=I$. For a $2\times2$ matrix

provided $\det A = ad-bc \neq 0$. If $\det A=0$ the matrix is singular (no inverse). To solve $A\mathbf{x}=\mathbf{b}$, use $\mathbf{x}=A^{-1}\mathbf{b}$.

$2\times2$: $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix}=ad-bc$.

$3\times3$ (expansion along row 1):

Cramer's Rule: for $A\mathbf{x}=\mathbf{b}$ with $D=\det A\neq0$, each variable is $x_k=D_k/D$, where $D_k$ replaces column $k$ of $A$ with $\mathbf{b}$.

For a proper fraction (numerator degree $<$ denominator degree), factor the denominator and write one term per factor:

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• Distinct linear $(ax+b)$: term $\dfrac{A}{ax+b}$.
• Repeated linear $(ax+b)^2$: terms $\dfrac{A}{ax+b}+\dfrac{B}{(ax+b)^2}$.
• Irreducible quadratic $(ax^2+bx+c)$: term $\dfrac{Ax+B}{ax^2+bx+c}$.

If improper, do polynomial division first. Solve for the constants by clearing denominators.

Graph each inequality and shade the overlap---the feasible region. In linear programming you maximize or minimize a linear objective $z=ax+by$ over that region. The optimum (if it exists) always occurs at a corner (vertex), so evaluate $z$ at every vertex and pick the best.

Every square matrix satisfies its own characteristic equation. For a $2\times2$ matrix $A$ with trace $t=\operatorname{tr}A=a+d$ and determinant $\Delta=\det A$, the Cayley--Hamilton theorem gives

Two powerful consequences:

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• Reduce high powers: $A^2$ (and hence $A^3, A^4,\dots$) is always a linear combination $\alpha A + \beta I$. So $A^n = c_n A + d_n I$ for scalars found by a recurrence, never a full multiplication.
• Instant inverse: rearrange to $A\bigl(tI-A\bigr)=\Delta I$, so if $\Delta\neq0$, @@BLOCK1@@

The scalar recurrence itself mirrors the matrix: if $A^n=c_nA+d_nI$ then $A^{n+1}=c_n A^2 + d_n A = (tc_n+d_n)A - \Delta c_n I$, so $c_{n+1}=tc_n+d_n$ and $d_{n+1}=-\Delta c_n$.

Some determinants have closed forms worth memorizing:

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• Vandermonde: $\det\begin{bmatrix} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ 1 & x_3 & x_3^2 \end{bmatrix}=(x_2-x_1)(x_3-x_1)(x_3-x_2)$, i.e. the product of all differences $x_j-x_i$ ($i<j$).
• Identity plus all-ones ($I+aJ$): if $J$ is the $n\times n$ matrix of all $1$'s, then $\det(I+aJ)=1+na$.
• Block triangular: $\det\begin{bmatrix} A & B \\ \mathbf{0} & D \end{bmatrix}=\det A\cdot\det D$ (the off-diagonal block $B$ does not matter).

Singular ($\det=0$) exactly when rows/columns are linearly dependent: one row is a scalar multiple of another, a row is a sum of others, or a row (or column) is all zeros. A singular matrix has no inverse, and $A\mathbf{x}=\mathbf{b}$ then has either no solution or infinitely many.

When both equations are conics, eliminate the squared terms rather than a single variable. Add or subtract multiples so that $x^2$ (or $y^2$) cancels, leaving a simpler relation; substitute back. Two conics can meet in up to four points, so expect several ordered pairs and pair each recovered value with the correct sign from the equation you used.

Two cases beyond the distinct-linear setup demand extra terms:

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• Repeated linear $(x-r)^k$: include one term per power, $\dfrac{A_1}{x-r}+\dfrac{A_2}{(x-r)^2}+\cdots+\dfrac{A_k}{(x-r)^k}$.
• Irreducible quadratic $x^2+bx+c$ (no real roots): use a linear numerator $\dfrac{Bx+C}{x^2+bx+c}$.

Root-substitution alone will not find every constant here, so clear denominators and match coefficients of like powers of $x$, giving a small linear system for the unknowns.

The corner principle assumes a bounded feasible region. Two degenerate cases:

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• Unbounded region: if the objective can grow without limit along an open direction, there is no maximum (though a minimum may still exist at a vertex, or vice versa). Always check whether the region is closed in the direction you are optimizing.
• Edge of optima: when the objective line $z=ax+by$ is parallel to a boundary edge, every point on that edge is optimal --- infinitely many solutions, all giving the same $z$. You detect this when two adjacent vertices tie for the best value.