Chapter 3 — Matrices

What is a matrix? A matrix is a rectangular array of numbers (or functions, symbols) arranged in rows and columns. An m × n matrix has m rows and n columns. Entries are denoted by a_{ij} where i is the row number and j is the column number.

Order of a matrix: The order (or size) of a matrix is written as m × n (m rows, n columns). Example: a 2 × 3 matrix has 2 rows and 3 columns.

Equality of matrices: Two matrices A and B are equal (A = B) iff they have the same order and all corresponding entries are equal: a_{ij} = b_{ij} for every i,j.

Types of matrices (common):

• Row matrix: 1 × n (one row).
• Column matrix: m × 1 (one column).
• Square matrix: n × n (same number of rows and columns).
• Zero (null) matrix: all entries 0, written O.
• Diagonal matrix: square with nonzero entries only on the main diagonal.
• Scalar matrix: a diagonal matrix with equal diagonal entries (λI).
• Identity matrix: I_n, diagonal entries 1; acts as multiplicative identity.
• Upper / lower triangular: entries below (or above) diagonal are zero.
• Symmetric: A^T = A (a_{ij} = a_{ji}).
• Skew-symmetric: A^T = −A (diagonal entries are 0).

Basic operations:

• Addition: A + B is defined only when A and B have same order. (A + B)_{ij} = a_{ij} + b_{ij}.
• Scalar multiplication: kA multiplies each entry by scalar k. (kA)_{ij} = k a_{ij}.
• Matrix multiplication: If A is m × n and B is n × p, then AB is m × p with (AB)_{ij} = Σ_{k=1}^{n} a_{ik} b_{kj}. Note: AB need not equal BA; multiplication is generally not commutative.
• Transpose: A^T is obtained by swapping rows and columns: (A^T)_{ij} = a_{ji}. Transpose converts m × n to n × m.
• Inverse (for square matrices): A^{-1} exists for an n × n matrix A iff there exists a matrix B with AB = BA = I_n. A must be non-singular (det(A) ≠ 0) to have an inverse.

Important properties (valid when sizes match):

• (A + B) + C = A + (B + C) (associativity of addition)
• A + B = B + A (commutativity of addition)
• A(BC) = (AB)C (associativity of multiplication)
• A(B + C) = AB + AC and (A + B)C = AC + BC (distributivity)
• (A^T)^T = A, (A + B)^T = A^T + B^T, (kA)^T = k A^T
• (AB)^T = B^T A^T (note the reverse order)
• If A is invertible, (A^{-1})^{-1} = A and (AB)^{-1} = B^{-1} A^{-1}
• Trace: tr(A) = Σ_{i} a_{ii} (defined for square matrices). tr(AB) = tr(BA) when both products are defined.

Interpretation: Matrices represent linear data arrays, linear maps and transformations. An n × n matrix acts on n-dimensional vectors; multiplication by A transforms vectors (geometry) and encodes systems of linear equations (algebra).

How this helps solve problems: Representing coefficients of linear systems as an augmented matrix simplifies Gaussian elimination. Matrices compactly represent relations (graphs, networks), transformations (rotations, scalings in geometry) and data tables (statistics, economics).

Introduction
A matrix is a rectangular array of numbers arranged in rows and columns. In Class 12 (CBSE) we classify matrices by their shape and by special algebraic properties. Knowing types helps in solving systems, representing networks, images and linear transformations.

Common Types (definitions and key properties)

• Row matrix: has a single row. Example: [a1 a2 ... an].
• Column matrix: has a single column. Example: \([a1; a2; ...; am]\).
• Zero (null) matrix: all entries are 0. Denoted O. Property: A + O = A.
• Square matrix: number of rows = number of columns (n × n). Many special matrices are square.
• Rectangular matrix: rows ≠ columns (m × n with m ≠ n).
• Diagonal matrix: square with nonzero entries only on the main diagonal. Off-diagonal entries are 0. Example: diag(d1,...,dn).
• Scalar matrix: a diagonal matrix with all diagonal entries equal to the same scalar λ; equals λI.
• Identity (unit) matrix: diagonal with 1s on the main diagonal, denoted I_n. Acts as multiplicative identity: AI_n = I_nA = A.
• Upper (lower) triangular matrix: square matrix with all entries below (above) the main diagonal equal to 0. Determinant = product of diagonal entries.
• Symmetric matrix: A = A^T. So a_ij = a_ji. Occurs for undirected graphs, covariance matrices.
• Skew-symmetric (antisymmetric) matrix: A^T = −A. Diagonal entries must be 0 (since a_ii = −a_ii ⇒ a_ii = 0).
• Orthogonal matrix: A^T = A⁻¹, equivalently A^TA = I. Columns (and rows) form an orthonormal set.
• Singular and non-singular matrices: A square matrix is non-singular (invertible) iff det(A) ≠ 0. If det(A) = 0 it is singular (no inverse).
• Equal matrices: same order and corresponding entries equal.

Remarks and useful consequences

• Transpose: A^T is obtained by swapping rows and columns; many types are defined via transpose (symmetric, skew-symmetric, orthogonal).
• Triangular matrices are easy for solving linear systems (forward/back substitution).
• Diagonal/Scalar/Identity matrices are simple to invert (if diagonal entries ≠ 0) and to exponentiate.

Short summary table (informal)
Row/Column/Rectangular describe shape. Square matrices allow diagonal, triangular, symmetric, skew-symmetric, scalar, identity, orthogonal, singular/non-singular descriptions.

Overview: A matrix is a rectangular array of numbers arranged in rows and columns. Operations on matrices let us combine and transform matrices in ways that model many practical problems (graphics, systems of equations, economics, image processing).

Basic operations:

• Addition and subtraction: Two matrices A and B can be added or subtracted only if they have the same order (same number of rows and columns). The operation is entry-wise: (A + B)_ij = A_ij + B_ij.
• Scalar multiplication: Multiply every entry of a matrix A by a scalar k: (kA)_ij = k · A_ij.
• Matrix multiplication: Product AB is defined only when A is of order m × n and B is of order n × p. The result AB is of order m × p with entries
  C = AB, where C_ij = Σ_k=1..n A_ik · B_kj.
• Transpose: The transpose of A (denoted A^T) swaps rows and columns: (A^T)_ij = A_ji. A is symmetric if A = A^T, skew-symmetric if A^T = −A.
• Identity and zero matrices: I_n is the n×n identity with 1s on the diagonal and 0s elsewhere. For any compatible A, AI = IA = A. The zero matrix O has all entries 0 and A + O = A.

Important properties (for compatible sizes):

• Associative (for multiplication): (AB)C = A(BC).
• Distributive: A(B + C) = AB + AC, (A + B)C = AC + BC.
• Transpose rules: (A + B)^T = A^T + B^T, (kA)^T = kA^T, (AB)^T = B^TA^T.
• Non-commutativity: In general AB ≠ BA (even when both products are defined).
• Inverse (square non-singular A): A⁻¹ defined by AA⁻¹ = A⁻¹A = I; (AB)⁻¹ = B⁻¹A⁻¹, and (A⁻¹)^T = (A^T)⁻¹.

Worked numeric examples:

Addition (A and B must be same order):

A = [[2,  -1],
     [0,   3]]
B = [[1,   4],
     [5,  -2]]

A + B = [[2+1,  -1+4],
         [0+5,   3+(-2)]] = [[3, 3], [5, 1]]

Multiplication (A is 2×3, B is 3×2):

A = [[1, 2, 0],
     [3, -1, 4]]   (2×3)
B = [[2, 1],
     [0, 3],
     [5, 2]]       (3×2)

AB is 2×2 with entries:
(AB)_{11} = 1*2 + 2*0 + 0*5 = 2
(AB)_{12} = 1*1 + 2*3 + 0*2 = 7
(AB)_{21} = 3*2 + (-1)*0 + 4*5 = 6 + 0 + 20 = 26
(AB)_{22} = 3*1 + (-1)*3 + 4*2 = 3 -3 +8 = 8

AB = [[2, 7], [26, 8]]

Transpose and symmetry example:

A = [[1, 4, -2],
     [4, 0,  3],
     [-2, 3, 5]]

A^T = A (so A is symmetric)

Tips/Checks:

• Always check dimensions before adding or multiplying.
• For multiplication think: rows of A dot columns of B.
• Use identity matrix to check inverses; use transpose rules when manipulating expressions.

Transpose (A^T)
For any m×n matrix A = [a_ij], the transpose A^T is the n×m matrix obtained by interchanging rows and columns: (A^T)_ij = a_ji. Geometrically, taking a transpose reflects the matrix entries across the main diagonal.

Symmetric matrix
A square matrix A (n×n) is symmetric if A^T = A. Equivalently, a_ij = a_ji for all i,j. The main diagonal entries may be any real numbers and off-diagonal entries appear in equal pairs.

Skew-symmetric (antisymmetric) matrix
A square matrix A is skew-symmetric if A^T = −A. Equivalently, a_ij = −a_ji for all i,j. In particular, all diagonal entries satisfy a_ii = 0.

Important facts and consequences

• Basic transpose properties: (A^T)^T = A, (A + B)^T = A^T + B^T, (kA)^T = kA^T, and (AB)^T = B^TA^T.
• Every square matrix A can be written uniquely as the sum of a symmetric and a skew-symmetric matrix:
  A = S + K where S = (A + A^T)/2 (symmetric) and K = (A − A^T)/2 (skew-symmetric).
• Diagonal entries of any skew-symmetric matrix are zero.
• Symmetric matrices often arise from inner products and covariances; skew-symmetric matrices appear in representations of cross products and infinitesimal rotations.

How to check quickly

• Compute A^T and compare to A. If equal → symmetric. If equal to −A → skew-symmetric. If neither, it is neither.
• To decompose, compute S = (A + A^T)/2 and K = (A − A^T)/2 and verify A = S + K.

Short example (illustrative calculation)
Let A = [[2, 3], [3, 5]]. Then A^T = [[2, 3], [3, 5]] = A, so A is symmetric.
Let B = [[0, 4], [−4, 0]]. Then B^T = [[0, −4], [4, 0]] = −B, so B is skew-symmetric.

Overview. Elementary row and column operations are simple manipulations of rows or columns of a matrix used to simplify matrices and solve linear systems. Echelon forms (row echelon form and reduced row echelon form) are canonical forms reached by these operations that make the structure of a system — pivots, rank and free variables — explicit.

Elementary operations (row version). There are three types of elementary row operations on an m × n matrix A:

• R1: Row swap — interchange two rows, Ri ↔ Rj.
• R2: Row scaling — multiply a row by a nonzero scalar k, k·Ri.
• R3: Row replacement — add a scalar multiple of one row to another, Ri → Ri + k·Rj.

Column operations C1, C2, C3 are analogous (operate on columns). Row operations preserve the solution set of a linear system represented by the augmented matrix (except column operations generally change the solution set unless applied to both coefficient matrix and constants in a compatible way).

Effect on determinant (for square matrices):

• Swapping two rows multiplies determinant by −1.
• Multiplying a row by scalar k multiplies determinant by k.
• Adding a scalar multiple of one row to another leaves determinant unchanged.

Elementary matrices. Each elementary row operation corresponds to left-multiplication by an elementary matrix E (invertible). If E is elementary, then performing that operation on A is equivalent to EA. Similarly, column operations correspond to right multiplication by elementary matrices.

Echelon forms.

• Row echelon form (REF):
  • All nonzero rows are above any zero rows.
  • The leading entry (first nonzero entry from left) of each nonzero row is to the right of the leading entry of the row above it.
  • Entries below each leading entry are zero.
• Reduced row echelon form (RREF):
  • It is in REF, and each leading entry is 1.
  • Each leading 1 is the only nonzero entry in its column (zeros both below and above).
  • RREF is unique for a given matrix; REF is not unique.

Pivots, rank and solutions. Pivot positions (leading entries) determine the rank. rank(A) = number of pivots = dimension of column space. For an m×n system Ax = b, after reducing the augmented matrix to RREF, you can read off whether the system is inconsistent (row [0 ... 0 | c] with c ≠ 0), has a unique solution (pivot in every variable column), or infinitely many solutions (free variables exist).

How to obtain REF/RREF (Gaussian and Gauss–Jordan elimination). Steps (row operations only):

1. Start with the leftmost nonzero column as the pivot column.
2. Swap rows to move a nonzero entry into the pivot position if needed.
3. Scale the pivot row to make pivot = 1 (for RREF) or leave as nonzero (for REF).
4. Use row replacement to make entries below (Gaussian) and above (Gauss–Jordan) the pivot zero.
5. Repeat for the submatrix that lies below and to the right of the pivot.

Practical notes. Use row operations to solve linear systems, compute rank, find inverses (apply operations to [A | I] to get [I | A^{-1}] when A is invertible), and to simplify matrices for determinant computation (tracking determinant changes).

Real-life connection. Many applications reduce to linear systems: circuit analysis (Kirchhoff laws), balancing chemical equations, resource allocation, and least-squares fitting. Row operations and echelon forms are the standard tools to solve these systems efficiently.

Rank of a Matrix

Definition: The rank of a matrix A (denoted rank(A)) is the maximum number of linearly independent rows or columns of A. Equivalently, it is the dimension of the row space or the column space of A. Row-rank = column-rank, so the two definitions are consistent.

Why it matters: Rank tells how many independent directions (degrees of freedom) a matrix or the corresponding linear transformation has. It determines solvability and number of solutions of linear systems, invertibility of square matrices, and the dimension of image of a linear map.

Methods to find rank

• Row-reduction (recommended): Reduce A to its row-reduced echelon form (RREF) using elementary row operations. The number of nonzero rows in RREF = rank(A).
• Using minors: The rank is the largest integer r such that there exists at least one nonzero r×r minor (determinant of an r×r submatrix). If all (r+1)×(r+1) minors are zero but some r×r minor is nonzero, rank = r.

Small worked example (row reduction)

Find the rank of A =

[1  2  3
 2  4  6
 1  0 -1]

R2 -> R2 - 2R1  gives R2 = [0 0 0]
R3 -> R3 - R1   gives R3 = [0 -2 -4]

R2 = [0 -2 -4]  => R2 -> (-1/2)R2 = [0 1 2]
R1 -> R1 - 2R2  => R1 = [1 0 -1]

Important consequences

• If A is n×n, A is invertible iff rank(A) = n (equivalently det(A) ≠ 0).
• Rank(A) ≤ min(number of rows, number of columns).
• Rank(A) = Rank(A^T).
• Rank(AB) ≤ min(rank(A), rank(B)).
• Rank-nullity theorem (for an m×n matrix viewed as linear map R^n → R^m): rank(A) + nullity(A) = n, where nullity(A) is the dimension of the solution space of Ax = 0.
• Rouché–Capelli theorem: A linear system Ax = b is consistent iff rank(A) = rank([A|b]). If consistent, number of free variables = n - rank(A).

Tips: Use row reduction for numerical/school problems. Use minors when a theoretical bound or exact minor is easier to compute. For large numerical matrices, singular value decomposition (SVD) gives numerical rank via nonzero singular values (useful in data applications).

Definition: For a square matrix A of order n, a matrix B of the same order is called an inverse of A if AB = BA = I_n, where I_n is the n×n identity matrix. If such B exists, A is called invertible (or non-singular) and the inverse is denoted A^-1. The inverse (when it exists) is unique.

When does an inverse exist? An n×n matrix A has an inverse iff det(A) ≠ 0. If det(A) = 0, A is singular and non-invertible.

Key properties:

• Uniqueness: if B and C both satisfy AB = BA = I, then B = C.
• Inverse of a product: (AB)^-1 = B^-1A^-1 (order reverses).
• Inverse of a transpose: (A^T)^-1 = (A^-1)^T.
• Scalar multiple: if k ≠ 0, (kA)^-1 = (1/k) A^-1.
• Inverse of inverse: (A^-1)^-1 = A.
• Special matrices: For an orthogonal matrix Q, Q^-1 = Q^T. For a diagonal matrix D with nonzero diagonal entries d_i, D^-1 is diagonal with entries 1/d_i.

How to compute the inverse:

• 2×2 formula (fast): For A = [[a, b], [c, d]], if det(A)=ad-bc ≠ 0 then A^-1 = (1/(ad-bc)) [[d, -b], [-c, a]].
• Adjoint (classical) method for n×n: A^-1 = (1/det(A)) adj(A), where adj(A) is the transpose of the cofactor matrix. Steps: compute all cofactors C_ij, form cofactor matrix [C_ij], transpose it to get adj(A), then divide by det(A).
• Gauss–Jordan elimination (recommended for computation): Augment A with I: [A | I]. Apply row operations to transform A to I. The transformed right block becomes A^-1: [I | A^-1]. If you cannot reach I (a zero row appears on left), A is singular.

Notes on non-square matrices: Non-square matrices do not have two-sided inverses. They may have a left-inverse or right-inverse if rank conditions permit. For general least-squares or non-square cases, one uses the Moore–Penrose pseudo-inverse A⁺.

Example of use: Finding A^-1 lets you solve linear systems Ax = b by x = A^-1b (when A is invertible). In many applications the inverse is not computed explicitly; instead numerical methods solve the system directly for stability and efficiency.

Common pitfalls: Do not attempt to compute inverse when det(A)=0. For numerical work, computing explicit inverses can be unstable; prefer solving linear systems via decomposition (LU, QR) when possible.

Overview: A system of linear equations is a collection of linear equations involving the same set of variables. In matrix language a system is written as A x = b, where A is the coefficient matrix, x is the column vector of unknowns and b is the constant column vector. Matrices provide systematic methods to find solutions and to determine whether solutions exist.

Key concepts:

• Augmented matrix: Formed by concatenating A and b as [A | b]. Row operations on [A | b] are used in elimination methods.
• Rank: The rank of a matrix is the maximum number of linearly independent rows (or columns). Denote rank(A) and rank([A | b]).
• Rouché-Capelli theorem: The system A x = b is consistent (has at least one solution) iff rank(A) = rank([A | b]). If consistent, then
  • unique solution when rank(A) = number of unknowns n,
  • infinitely many solutions when rank(A) < n.
• Homogeneous system A x = 0 is always consistent (x = 0). If rank(A) < n it has nontrivial solutions.

Matrices methods to solve systems:

• Inverse method: If A is square and invertible (det(A) != 0), then the unique solution is x = A^{-1} b. This is direct but requires computing A^{-1}.
• Gaussian elimination / Row reduction: Use elementary row operations on the augmented matrix [A | b] to reach row echelon or reduced row echelon form. Read solutions by back-substitution or from reduced form. This method also determines rank and consistency.
• Cramer’s rule (for square systems with det(A) != 0): each variable x_i = det(A_i)/det(A), where A_i is A with column i replaced by b. Useful for theoretical purposes and small systems only.

Elementary row operations (preserve solution set):

• Swap two rows,
• Multiply a row by a nonzero constant,
• Add a multiple of one row to another row.

Interpreting solutions geometrically:

• Two variables: each linear equation is a line in the plane. Unique solution = intersection point; infinite solutions = coincident lines; no solution = parallel distinct lines.
• Three variables: each equation is a plane in 3D. Solutions correspond to intersection of planes: point, line, plane (infinitely many) or empty (no common intersection).

How to check consistency quickly:

• Form [A | b] and perform row reduction until you can read ranks (count nonzero rows in echelon form).
• Apply Rouché-Capelli: if rank(A) = rank([A | b]) system is consistent; otherwise inconsistent.

Tip: For computational work use Gaussian elimination (stable and systematic). For small square systems, inverse or Cramer’s rule is fine.

Definition: A set of vectors {v1, v2, ..., vk} in R^n is linearly dependent if there exist scalars c1, c2, ..., ck, not all zero, such that c1 v1 + c2 v2 + ... + ck vk = 0 (the zero vector). If the only solution is c1 = c2 = ... = ck = 0, the set is linearly independent.

Understanding the idea:

• Linear dependence means at least one vector in the set can be expressed as a linear combination of the others.
• Linear independence means no vector in the set can be written using the others; each adds a new direction.

Matrix viewpoint: Put vectors as columns of a matrix A = [v1 v2 ... vk]. Then the relation c1 v1 + ... + ck vk = 0 is A c = 0, where c = (c1,...,ck)^T. The columns are linearly independent iff the only solution of A c = 0 is the trivial solution c = 0.

Tests and consequences:

• Rank test: Columns of A are independent iff rank(A) = number of columns. If rank < number of columns the columns are dependent.
• Square matrix (n × n): columns are independent iff det(A) ≠ 0.
• Geometric meaning in low dimensions:
  • In R^2: two vectors are independent iff they are not scalar multiples (not collinear).
  • In R^3: three vectors are independent iff they are not coplanar (they span 3D space).
• Independence is required for a set to be a basis. The maximum size of an independent set in R^n is n (dimension).

Worked idea (small check): For vectors v1 = (1,2), v2 = (2,4) in R^2, check c1 v1 + c2 v2 = 0. This gives two scalar equations: c1 + 2 c2 = 0 and 2 c1 + 4 c2 = 0; these are the same equation, so there is a nontrivial solution (for example c1 = 2, c2 = -1). Hence v1 and v2 are linearly dependent (v2 = 2 v1).

Why it matters: Linear independence tells us whether vectors provide new information/direction. In solving systems, it tells if variables are uniquely determined. In data and signal processing, independence relates to redundancy and uniqueness of features.

Overview: A matrix is a rectangular array of numbers. Class 12 studies algebraic rules (properties) and important identities that matrices satisfy. These rules help manipulate matrix expressions, solve linear systems, and understand linear transformations.

Basic operations and definitions

• Zero matrix O: all entries 0. Identity matrix I: square matrix with 1s on diagonal, 0 elsewhere.
• Addition and scalar multiplication: defined entrywise. Product AB defined when number of columns of A equals number of rows of B.

Key properties (with brief reasoning)

• Addition: A + B = B + A (commutative), (A + B) + C = A + (B + C) (associative).
• Scalar distributivity: k(A + B) = kA + kB and (k + l)A = kA + lA.
• Multiplication associative: A(BC) = (AB)C when sizes conform. This follows from combining linear combinations in order.
• Multiplication not generally commutative: AB and BA may both exist but usually AB ≠ BA. Exception: special matrices that commute.
• Distributive laws: A(B + C) = AB + AC and (A + B)C = AC + BC.
• Identity and zero: AI = IA = A (if sizes conform). A0 = 0 and 0A = 0.

Transpose properties

• (A^T)^T = A.
• (A + B)^T = A^T + B^T.
• (kA)^T = k A^T for scalar k.
• Important identity: (AB)^T = B^T A^T (order reverses).

Inverse properties (for square matrices)

• If A is invertible (non-singular), there exists A^{-1} with AA^{-1} = A^{-1}A = I.
• Product inverse: (AB)^{-1} = B^{-1} A^{-1} when A and B are invertible.
• Transpose of inverse: (A^{-1})^T = (A^T)^{-1}.
• Determinant: det(AB) = det(A) det(B). If det(A) ≠ 0 then A is invertible.

Special matrices and identities

• Symmetric: A^T = A. Real symmetric matrices represent e.g. covariance matrices.
• Skew-symmetric: A^T = −A. Diagonals are zero.
• Orthogonal: A^T = A^{-1}. Orthogonal matrices preserve lengths and angles (rotations, reflections).
• Idempotent: A^2 = A (projections). Geometric meaning: applying transformation twice same as once.
• Involutory: A^2 = I (self-inverse transformations like 180° rotation in some contexts).
• Nilpotent: A^k = O for some k ≥ 1 (eventually sends vectors to zero).
• Decomposition: Every square matrix A can be written as sum of symmetric and skew-symmetric parts: A = (1/2)(A + A^T) + (1/2)(A - A^T).

Other useful identities

• trace(AB) = trace(BA) for square matrices (useful in proofs, not implying AB = BA).
• adj(AB) = adj(B) adj(A) (adjoint/rearrangement identity) when A and B are square.

Why these matter (applications): Properties let you simplify expressions, compute inverses of products, analyze linear maps (rotations, projections), and use matrices in probability (Markov chains), networks (adjacency matrices), computer graphics (transformations), statistics (covariance matrices), and engineering.

Overview
In Class 12 Matrices, the main applications revolve around solving systems of linear equations, studying consistency of systems, homogeneous vs non-homogeneous systems, and modelling simple real-world situations (allocation/mixing, transformations, Markov processes). Matrices provide compact notation and efficient methods (Cramer's rule, inverse of a matrix, row reduction/rank) to obtain solutions and to determine whether solutions exist and whether they are unique.

Core ideas and procedures

• Representing a system: A linear system of m equations in n unknowns can be written as A x = b, where A is an m×n coefficient matrix, x is an n×1 column vector of unknowns and b is an m×1 constant vector.
• Solving when A is square and invertible (n×n): If det(A) ≠ 0 then the unique solution is x = A^-1 b.
• Cramer's rule (square systems): If det(A) ≠ 0 then each unknown x_i = det(A_i)/det(A), where A_i is A with its i-th column replaced by b.
• Rank and consistency (Rouché–Capelli theorem): Form the augmented matrix [A|b]. The system A x = b is consistent iff rank(A) = rank([A|b]). If consistent and rank(A) = n (number of unknowns) the solution is unique; if rank(A) < n there are infinitely many solutions (parameters appear).
• Homogeneous systems: A x = 0 is always consistent (x = 0). It has non-trivial solutions iff rank(A) < n. For an n×n matrix A, a non-trivial solution exists iff det(A) = 0.
• Finding A^-1: For a square matrix, compute A^-1 using adjoint/determinant (for small matrices) or elementary row operations (augment with I and row-reduce).

Typical problem types & strategies

• Solve a system of 2 or 3 linear equations: try matrix inverse if det ≠ 0, otherwise use row reduction to find general solution.
• Determine parametric values for which a system is consistent/inconsistent: compute rank(A) and rank([A|b]) as functions of the parameter and compare.
• Find conditions for non-trivial solutions of homogeneous systems: set det(A)=0 (for square A) or check rank < n.
• Modeling problems: translate real-world constraints into linear equations, form A and b, then solve by appropriate method.

Short solution workflow

1. Translate the word problem into linear equations and form A, x, b.
2. If A is square compute det(A). If det(A) ≠ 0 use x = A^-1b or Cramer's rule.
3. If det(A) = 0 or A is rectangular, compute ranks of A and [A|b] by row-reduction to decide consistency and find general solution (introduce parameters for free variables).
4. Interpret the solution in context (e.g., non-negative constraints for real-life quantities).

Practical tips

• Use row-reduction (Gaussian elimination) to find rank and solutions; it works for square and rectangular matrices.
• For 2×2 matrices use the closed-form inverse and Cramer's rule for quick calculation.
• Always check the applicability conditions: Cramer's rule and A^-1 require det(A) ≠ 0.