Matrices

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

The determinant is a scalar value that is a function of the entries of a square matrix and provides important properties of the matrix, such as whether it is invertible.

To find the inverse of a matrix A, you can use the formula A^(-1) = adj(A)/det(A), where adj(A) is the adjugate of A and det(A) is the determinant of A.

A matrix has no inverse if its determinant is zero, indicating that it is singular and does not span the entire space.

Cramer's Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, using determinants.

Matrix addition is performed by adding corresponding elements of two matrices of the same dimensions.

Matrix subtraction is the operation of subtracting corresponding elements of two matrices of the same dimensions.

The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, denoted as I, and it acts as the multiplicative identity in matrix multiplication.

The transpose of a matrix is obtained by swapping its rows and columns, denoted as A^T.

A square matrix is a matrix with the same number of rows and columns.