MATRICES

Matrix

A rectangular array of numbers or variables arranged in rows and columns and enclosed in brackets or parentheses.

Order of a Matrix

The size of a matrix written as m x n where m is the number of rows and n is the number of columns.

Entry (Element) of a Matrix

An individual value inside a matrix usually denoted as Aij where i is the row and j is the column.

Coefficient

Another term for the entries or elements of a matrix.

Column Matrix

A matrix that has only one column.

Row Matrix

A matrix that has only one row.

Square Matrix

A matrix with the same number of rows and columns.

Zero Matrix

A matrix in which all entries are equal to zero.

Matrix Equality

Two matrices are equal if they have the same order and all corresponding entries are equal.

Matrix Addition

An operation where corresponding entries of matrices with the same dimensions are added together.

Matrix Subtraction

An operation where corresponding entries of matrices with the same dimensions are subtracted.

Scalar Multiplication

The multiplication of every entry of a matrix by a single number.

Matrix Multiplication

The operation of multiplying two matrices where the number of columns of the first matrix equals the number of rows of the second matrix.

Compatibility Condition for Multiplication

The requirement that the inner dimensions of two matrices must match in order for multiplication to be possible.

Resulting Matrix Order (Multiplication)

If A is m x n and B is n x p then AB is m x p.

Row-by-Column Rule

The method of multiplying matrices by multiplying a row of the first matrix with a column of the second matrix and adding the results.

Transpose of a Matrix

A matrix obtained by interchanging the rows and columns of the original matrix.

Identity Matrix

A square matrix with ones on the main diagonal and zeros elsewhere that acts as the multiplicative identity in matrix multiplication.

Main Diagonal

The diagonal entries of a square matrix from the top-left corner to the bottom-right corner.

Determinant

A scalar value computed from a square matrix that is used to determine properties such as invertibility.

Determinant of a 2 x 2 Matrix

For matrix [[a b] [c d]] the determinant is ad minus bc.

Determinant of a 3 x 3 Matrix

A scalar value computed using expansion by minors across a row or column.

Minor of an Entry

The determinant of the matrix formed by removing the row and column of a specific entry.

Matrix of Minors

A matrix composed of the minors of each entry of the original matrix.

Cofactor

A minor multiplied by either positive one or negative one according to its position in the matrix.

Cofactor Matrix

The matrix formed by applying the sign pattern to the matrix of minors.

Adjoint (Adjugate) Matrix

The transpose of the cofactor matrix.

Inverse of a Matrix

A matrix A inverse that satisfies A multiplied by A inverse equals the identity matrix.

Invertible Matrix

A square matrix whose determinant is not equal to zero.

Singular Matrix

A square matrix whose determinant is equal to zero and therefore has no inverse.

Formula for Inverse (3 x 3)

A inverse equals one over the determinant of A multiplied by the adjoint of A.

System of Linear Equations

A collection of linear equations involving the same variables.

Matrix Equation

An equation written in the form A multiplied by V equals B.

Coefficient Matrix

The matrix containing the coefficients of the variables in a system of equations.

Variable Matrix

The matrix containing the variables of a system of equations.

Constant Matrix

The matrix containing the constant terms in a system of equations.

Solving System Using Inverse

If A multiplied by V equals B then V equals A inverse multiplied by B provided A is invertible.