Matrices, Linear Systems & Determinants – Core Vocabulary

A comprehensive set of vocabulary flashcards covering the foundational concepts, structures, and theorems related to matrices, linear systems, and determinants discussed in the lecture notes.

Matrix

A rectangular array of numbers (or other mathematical objects) arranged in m rows and n columns, denoted m × n.

Order (Size) of a Matrix

The pair of integers m × n that indicates the number of rows (m) and columns (n) in a matrix.

Entry (Element)

An individual number in a matrix located at row i and column j, written aᵢⱼ.

Row Matrix (Row Vector)

A 1 × n matrix consisting of a single horizontal row.

Column Matrix (Column Vector)

An n × 1 matrix consisting of a single vertical column.

Square Matrix

A matrix with the same number of rows and columns (n × n).

Principal Diagonal

The set of elements a₁₁, a₂₂, …, aₙₙ running from the top left to the bottom right of a square matrix.

Secondary Diagonal

The elements a₁ₙ, a₂,ₙ₋₁, …, aₙ₁ running from the top right to the bottom left of a square matrix.

Diagonal Matrix

A square matrix whose off-diagonal entries are all zero.

Zero (Null) Matrix

A matrix in which every entry is zero; acts as the additive identity.

Identity Matrix

A square diagonal matrix with 1’s on the principal diagonal and 0’s elsewhere; denoted I or Iₙ.

Upper Triangular Matrix

A square matrix whose entries below the principal diagonal are all zero.

Lower Triangular Matrix

A square matrix whose entries above the principal diagonal are all zero.

Symmetric Matrix

A square matrix that satisfies A = Aᵗ, meaning aᵢⱼ = aⱼᵢ.

Skew-Symmetric (Antisymmetric) Matrix

A square matrix that satisfies A = −Aᵗ; in particular, all diagonal elements are zero.

Transpose

The matrix obtained by interchanging the rows and columns of a given matrix, denoted Aᵗ or Aᵀ.

Opposite Matrix

The matrix −A whose entries are the negatives of those in A; satisfies A + (−A)=0.

Scalar Multiplication (of a Matrix)

The operation kA produced by multiplying every entry of matrix A by scalar k.

Matrix Addition

The entry-wise sum of two matrices of the same size, (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ.

Matrix Product

For A (m × r) and B (r × n), the m × n matrix AB with entries (AB)ᵢⱼ = ∑_{k=1}^r aᵢₖ bₖⱼ.

Conformability Condition

The requirement that the number of columns of A equals the number of rows of B for AB to be defined.

Rank (Posto)

The number of non-zero rows in a matrix’s reduced row echelon form; equals the dimension of its row (or column) space.

Nullity

For an m × n matrix, n minus the rank; equals the dimension of the solution space of AX = 0.

Row Echelon Form (REF)

A staircase-shaped matrix form where each non-zero row starts to the right of the row above and zero rows are at the bottom.

Reduced Row Echelon Form (RREF)

An REF in which every leading 1 is the only non-zero entry in its column.

Elementary Row Operations

(1) Row swap, (2) Multiply a row by non-zero scalar, (3) Add scalar multiple of one row to another; preserve row equivalence.

Pivot

The first non-zero entry in a non-zero row of an echelon form, usually scaled to 1.

System of Linear Equations

A collection of linear equations in the same variables, often written AX = B.

Homogeneous System

A linear system with all zero constant terms (B = 0); always has the trivial solution X = 0.

Equivalent Systems

Two systems that possess exactly the same solution set.

Augmented Matrix

The matrix [A | B] formed by appending the column of constants B to the coefficient matrix A.

Determinant

A scalar value associated with a square matrix, denoted det(A), that encodes area/volume scaling and invertibility.

Minor

The determinant of the submatrix obtained by deleting a specified row and column from a square matrix.

Cofactor

The signed minor Δᵢⱼ = (−1)^{i+j} × minor Aᵢⱼ; used in determinant expansion.

Adjugate (Adjoint) Matrix

The transpose of the cofactor matrix; denoted adj(A).

Invertible (Nonsingular) Matrix

A square matrix A with an inverse A⁻¹ satisfying AA⁻¹ = A⁻¹A = I; equivalently det(A) ≠ 0.

Inverse Matrix

The unique matrix A⁻¹ such that AA⁻¹ = A⁻¹A = I; given by adj(A)/det(A) when det(A) ≠ 0.

Cramer's Rule

Method for solving n × n systems AX = B when det(A) ≠ 0: xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A with B.

Sarrus Rule

Shortcut for computing determinants of 3 × 3 matrices using sums of products along diagonals.

Laplace Expansion

Formula expressing det(A) as a sum of cofactors times their corresponding entries along any row or column.

Triangular Determinant Property

The determinant of an upper or lower triangular matrix equals the product of its diagonal entries.

Determinant Row-Operation Properties

Swapping rows changes sign; scaling a row by k multiplies det by k; adding a multiple of one row to another leaves det unchanged.