Chapter Review

Matrices and Determinants

Matrices and Determinants · System of Linear Equations

Matrix Structure and Operations

Matrices are rectangular arrays with order $m \times n$. Addition requires matching dimensions and is element-wise, while scalar multiplication distributes over both matrix and scalar addition.

Key Points

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Element $a_{ij}$ is in row $i$, column $j$; addition/subtraction require identical dimensions
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Addition is commutative ($A + B = B + A$), but multiplication is NOT
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The null matrix $O$ is the additive identity: $A + O = A$
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Scalar distribution: $k(A + B) = kA + kB$ and $(k + l)A = kA + lA$

Matrix Multiplication

Matrix multiplication uses the dot product of rows and columns, not element-wise pairing. The inner dimensions must match for the product to be defined.

Key Points

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$(m \times n)(n \times p)$ produces an $(m \times p)$ matrix — inner $n$ values must agree
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Each entry: $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$
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Generally non-commutative: $AB \neq BA$
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Identity property: $AI = IA = A$
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Transpose of a product reverses order: $(AB)^T = B^T A^T$

Formula

$$c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$$

Special Matrix Types

Matrices are classified by structural properties of their entries. Symmetric, skew-symmetric, Hermitian, diagonal, scalar, and null matrices each have distinct algebraic behaviours.

Key Points

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Symmetric: $A^T = A$, i.e. $a_{ij} = a_{ji}$. All diagonal matrices are symmetric
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Skew-Symmetric: $A^T = -A$, forcing all diagonal entries to zero
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Hermitian: $A^\theta = A$ (complex analog of symmetric); diagonal entries must be real
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Skew-Hermitian: $A^\theta = -A$; diagonal entries are zero or purely imaginary
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Scalar matrix: $kI$ — commutes with all square matrices of the same order

Determinants

The determinant is a scalar value of a square matrix that determines invertibility. A zero determinant means the matrix is singular and has no inverse.

Key Points

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2×2: $|A| = ad - bc$
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3×3: expand along any row/column using cofactors $A_{ij} = (-1)^{i+j} M_{ij}$
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Choose the row/column with the most zeros for efficient expansion
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$|A| = 0$ means the matrix is singular (no inverse exists)
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Product rule: $|AB| = |A||B|$

Formula

$$A_{ij} = (-1)^{i+j} M_{ij}$$

Properties of Determinants

Several shortcut properties allow evaluating or simplifying determinants without full cofactor expansion, by recognising special structures.

Key Points

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$|A| = |A^T|$ — transpose does not change the determinant
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Swapping two rows/columns reverses the sign
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Two identical rows/columns or an all-zero row/column → $|A| = 0$
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Scaling one row by $k$ multiplies the determinant by $k$; scaling the whole $n \times n$ matrix: $|kA| = k^n|A|$
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Adding a multiple of one row to another leaves the determinant unchanged
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Triangular/diagonal matrix: determinant = product of diagonal entries

Formula

$$|kA| = k^n|A|$$

Adjoint and Matrix Inverse

The inverse $A^{-1}$ exists only when $|A| \neq 0$ and is computed via the adjoint matrix (transpose of the cofactor matrix) scaled by the reciprocal of the determinant.

Key Points

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$A^{-1} = \frac{1}{|A|} \text{adj}(A)$, requires $|A| \neq 0$
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2×2 adjoint shortcut: swap diagonal entries, negate off-diagonal entries
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$|A^{-1}| = \frac{1}{|A|}$
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Alternative: augment $[A | I]$ and row-reduce to $[I | A^{-1}]$
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Verification: $AA^{-1}$ must equal $I$

Formula

$$A^{-1} = \frac{1}{|A|} \text{adj}(A)$$

Systems of Linear Equations and Consistency

A linear system $AX = B$ is consistent if it has at least one solution. Comparing the rank of the coefficient matrix to the augmented matrix determines consistency and solution type.

Key Points

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Unique solution: rank($A$) = rank($[A|B]$) = number of variables
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Infinitely many: rank($A$) = rank($[A|B]$) < number of variables — free variables exist
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No solution: rank($A$) ≠ rank($[A|B]$) — contradiction row $[0 \ 0 \ 0 \ | \ k]$, $k \neq 0$
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Rank = number of non-zero rows after reducing to echelon form

Echelon Form and Row Reduction

Elementary row operations (swap, scale, row replacement) transform any augmented matrix into echelon form for back-substitution, or reduced echelon form for direct reading of solutions.

Key Points

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Three operations: row swap ($R_i \leftrightarrow R_j$), scaling ($kR_i$), row addition ($R_i + kR_j$)
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Echelon form: staircase of leading 1s, zeros below each pivot
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Reduced echelon form: additionally, zeros above each pivot — solution read directly
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All operations are reversible — they preserve the solution set
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Most general method: works even when $|A| = 0$

Cramer's Rule

Cramer's Rule solves each variable as a ratio of determinants — the numerator replaces the variable's column in $A$ with the constants vector $B$.

Key Points

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$x_i = \frac{|A_i|}{|A|}$ where $A_i$ has column $i$ replaced by $B$
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Requires $|A| \neq 0$ (non-singular system)
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For a 3×3 system, requires computing 4 determinants
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Replace a column, not a row, when forming $A_i$

Formula

$$x_i = \frac{|A_i|}{|A|}$$

Homogeneous Systems

A homogeneous system $AX = O$ always has the trivial solution $(0, 0, 0)$. Non-trivial solutions exist if and only if $|A| = 0$.

Key Points

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Trivial solution $X = O$ always satisfies $AX = O$
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$|A| = 0$: infinitely many solutions parameterised by free variables
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$|A| \neq 0$: only the trivial solution (inverse gives $X = A^{-1}O = O$)
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To find non-trivial solutions: set one variable as parameter $t$, solve remaining from any two equations, verify with the third

Formula

$$|A| = 0 \iff \text{non-trivial solutions exist}$$

Formulas

2×2 Determinant

Cross-multiply diagonals and subtract.

Formula

|A| = ad - bc

Cofactor Formula

Sign of a cofactor follows the checkerboard pattern.

Formula

$A_{ij} = (-1)^{i+j} M_{ij}$

Matrix Inverse

Inverse via adjoint scaled by reciprocal of determinant.

Formula

$A^{-1} = \frac{1}{|A|} \text{adj}(A)$

Scalar Determinant

Scaling an $n \times n$ matrix raises the scalar to the $n$-th power.

Formula

$|kA| = k^n |A|$

Product Determinant

Determinant of a product equals the product of determinants.

Formula

|AB| = |A||B|

Transpose Product Rule

Transpose of a product reverses the order of factors.

Formula

$(AB)^T = B^T A^T$

Cramer's Rule

Each variable is the ratio of a modified determinant to the original.

Formula

$x_i = \frac{|A_i|}{|A|}$

Matrix Inverse Solution

Direct solution for non-singular linear systems.

Formula

$X = A^{-1}B$