Matrices and Determinants Study Cards

This set of vocabulary flashcards covers fundamental concepts of matrix algebra and determinants, including matrix types, special matrix forms, operations like trace and transpose, and system of equations terminology.

Matrix

A set of $m \times n$ numbers (real or complex) arranged in the form of a rectangular array having $m$ rows and $n$ columns.

Singleton matrix

A matrix containing only one element, where the number of rows $m$ and columns $n$ are both equal to $1$.

Row matrix

A matrix having only one row, also called a row vector, with a general form of order $1 \times n$.

Column matrix

A matrix having only one column, also called a column vector, with a general form of order $m \times 1$.

Zero or Null matrix

A matrix in which all elements are equal to zero.

Horizontal matrix

A matrix of order $m \times n$ where the number of columns is greater than the number of rows (n > m).

Vertical matrix

A matrix of order $m \times n$ where the number of rows is greater than the number of columns (m > n).

Square matrix

A matrix where the number of rows is equal to the number of columns ($m = n$).

Conjugate elements

In a square matrix of order $n \times n$, the elements $a_{ij}$ and $a_{ji}$ are referred to by this term.

Diagonal Matrix

A square matrix in which all elements are zero except those in the principal diagonal ($a_{ij} = 0$ when $i \neq j$).

Scalar Matrix

A diagonal matrix in which all principal diagonal elements are equal to a constant value $K$.

Unit Matrix (Identity matrix)

A diagonal matrix in which each diagonal element is equal to unity ($1$), denoted as $I_n$.

Upper Triangular Matrix

A square matrix in which all elements below the principal diagonal are zero ($a_{ij} = 0$ for all i > j).

Lower Triangular Matrix

A square matrix in which all elements above the principal diagonal are zero ($a_{ij} = 0$ for all j > i).

Singular matrix

A square matrix whose determinant is equal to zero ($|A| = 0$).

Non-singular matrix

A square matrix whose determinant is not equal to zero ($|A| \neq 0$).

Submatrix

A matrix obtained by omitting any rows or columns of a given matrix.

Equality of Matrix

Two matrices are equal if they are of the same order and their corresponding elements are equal.

Trace of a matrix

The sum of the diagonal elements of a square matrix $A$, denoted as $tr(A)$, which is $\sum_{i=1}^n a_{ii}$.

Transpose of a matrix

The matrix obtained by interchanging the rows and columns of an original matrix $A$, denoted by $A^T$ or $A'$.

Orthogonal matrix

A square matrix $A$ such that the product with its transpose yields the identity matrix ($AA^T = A^T A = I$).

Idempotent matrix

A square matrix $A$ such that $A^2 = A$.

Involutory matrix

A square matrix $A$ such that $A^2 = I$; its determinant value is always $\pm 1$.

Nilpotent matrix

A square matrix $A$ such that $A^k = 0$ for some positive integer $k$; the least such value of $k$ is called the index.

Symmetric matrix

A square matrix $A$ that is equal to its transpose ($A = A^T$), meaning $a_{ij} = a_{ji}$ for all indices.

Skew-symmetric matrix

A square matrix $A$ such that $A^T = -1\times A$, meaning $a_{ij} = -a_{ji}$ and all diagonal elements are zero ($a_{ii} = 0$).

Minor

The value $M_{ij}$ obtained by deleting the $i^{th}$ row and $j^{th}$ column of a determinant.

Co-factor

The value of an element $a_{ij}$ defined as $C_{ij} = (-1)^{i+j} \times M_{ij}$, where $M_{ij}$ is the minor.

Adjoint of a matrix

The transpose of the matrix of co-factors of the elements of a given square matrix $A$, denoted by $adj(A)$.

Inverse of a Matrix

A matrix $A^{-1}$ such that $A \times A^{-1} = A^{-1} \times A = I$, calculated as $\frac{1}{|A|} \times adj(A)$, provided $|A| \neq 0$.

Consistent system

A system of equations that has one or more solutions.

Inconsistent system

A system of equations that has no solution.

Trivial solution

A solution to a homogeneous system where all variables are simultaneously zero ($x = y = z = 0$).

Characteristic Equation

The equation formed by taking the determinant of the matrix minus a scalar multiple of the identity matrix equal to zero ($|A - \lambda I| = 0$).