Matrices

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

Matrix addition involves adding corresponding elements of two matrices of the same dimensions.

Matrix multiplication involves taking the dot product of rows and columns of two matrices, where the number of columns in the first matrix must equal the number of rows in the second.

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

For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated as ad - bc.

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere, denoted as I.

The inverse of a matrix A is another matrix, denoted A^-1, such that when multiplied together, they yield the identity matrix: AA^-1 = I.

Matrix addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)).

A zero matrix is a matrix in which all elements are zero.

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