Finite Mathematics I - Matrix Reviewer Flashcards

Vocabulary flashcards covering core concepts, matrix types, properties, determinants, and elementary row operations from Finite Mathematics I.

Matrix

A rectangular arrangement of elements in rows and columns, usually enclosed in brackets and named with a capital letter, e.g., matrix $A$.

Element (or Entry)

Each individual number inside a matrix, where the element in row $i$ and column $j$ is denoted $a_{ij}$.

Order (or Dimension/Size)

Describes the number of rows and columns of a matrix, written as $m \times n$ (read "m by n"), where $m$ is the number of rows and $n$ is the number of columns.

Row

A horizontal line of elements in a matrix.

Column

A vertical line of elements in a matrix.

Principal (Main) Diagonal

The set of elements $a_{ij}$ where $i = j$, running from the top-left to the bottom-right corner of a square matrix.

Row Matrix

A matrix with only one row (order $1 \times n$).

Column Matrix

A matrix with only one column (order $m \times 1$).

Square Matrix

A matrix with the same number of rows and columns (order $n \times n$).

Rectangular Matrix

A matrix in which the number of rows is not equal to the number of columns.

Zero (Null) Matrix

A matrix in which every element is zero, denoted $O$.

Diagonal Matrix

A square matrix in which every element off the main diagonal is zero (diagonal entries may be zero or nonzero).

Scalar Matrix

A diagonal matrix in which all the main-diagonal elements are equal.

Identity Matrix

A scalar matrix whose main-diagonal elements are all $1$, denoted $I$, which acts like the number $1$ in matrix multiplication.

Symmetric Matrix

A square matrix that is equal to its own transpose ($A = A^T$).

Transpose of a Matrix

The matrix $A^T$ formed by interchanging the rows and columns of matrix $A$.

Equality of Matrices

The condition where two matrices $A$ and $B$ have the same order and their corresponding elements are equal ($a_{ij} = b_{ij}$ for every $i$ and $j$).

Determinant

A single real number associated with a square matrix that provides important information about the matrix, written as $\det(A)$ or $|A|$.

Cramer's Rule

A method that uses determinants to solve a system of linear equations directly, provided the determinant of the coefficient matrix is not zero.

Coefficient Matrix (D)

The determinant formed from the coefficients of the variables in a system of linear equations.

Elementary Row Operations

Operations performed on the rows of a matrix (often an augmented matrix representing a system of equations) that produce an equivalent system.

Augmented Matrix

A matrix formed by combining the coefficient matrix of a system with the column of constants, typically separated by a vertical line.

Equivalent Matrices

Matrices obtained from one another through elementary row operations; they represent systems with the same solution set.

Row Echelon Form (REF)

A matrix in which all nonzero rows are above any all-zero rows, and the leading entry of each row is to the right of the leading entry of the row above it.

Reduced Row Echelon Form (RREF)

A row echelon form matrix in which every leading entry is $1$, and each leading $1$ is the only nonzero entry in its column.

Pivot (Leading Entry)

The first nonzero number in a row, from left to right, used as the reference point for eliminating entries in the rows above or below it.

Row Interchange

An elementary row operation that swaps the positions of two rows, denoted $R_i \leftrightarrow R_j$.

Row Scaling

An elementary row operation that multiplies every element of a row by a nonzero constant $k$, denoted $kR_i \rightarrow R_i$.

Row Addition (Combination)

An elementary row operation that adds a multiple of one row to another row, replacing that row, denoted $R_i + kR_j \rightarrow R_i$.