Understanding Matrices

Matrices are arrays of numbers arranged in rows and columns, and are used in various branches of mathematics.
They have a range of properties and are manipulated according to specific rules.
Dimensions of a matrix are always defined by rows x columns.

Operations with Matrices

Matrix Addition/Subtraction: Matrices are added or subtracted element by element. The matrices must have the same dimensions to perform these operations.
Matrix Multiplication: To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The entries in the resulting matrix are calculated as the sum of the products of the corresponding row and column entries in the original matrices.
Scalar Multiplication: When a matrix is multiplied by a scalar, every entry in the matrix is multiplied by the scalar value.
Matrix Transposition: The transpose of a matrix is found by swapping its rows and columns.

Zero Matrix: A matrix where every element is zero.
Identity Matrix: A square matrix which has ones on the main diagonal from top left to bottom right and zeros everywhere else.
Diagonal Matrix: A matrix that has non-zero elements only on the main diagonal and zero elements everywhere else.
Square Matrix: A matrix which has the same number of rows and columns.

The determinant of a square matrix is a particular quantity that is used in system solving and matrix inversion. For a 2x2 matrix with entries a, b, c, d, the determinant is ad - bc.
The inverse of a square matrix A is a matrix that when multiplied by A, results in the identity matrix. Not all matrices have inverses, only those with non-zero determinants.

Matrices are used to solve systems of linear equations, represent geometric transformations, and model real-world problems in various scientific disciplines.