Matrices

By definition, for a 2×2 matrix: det = ad - bc.

A matrix is a rectangular array of numbers arranged in rows and columns. It is usually denoted by uppercase letters, such as A, B, C.

The matrix A here has 3 rows and 3 columns, so it is called a 3×3 matrix.

Matrix Operations

Addition: two matrices can be added if they are the same size.
Subtraction: similarly, subtract the values element-wise.
Scalar multiplication: multiply every element by the scalar.
Matrix multiplication: not element-wise, but row-column multiplication.

Determinant

The determinant is a number assigned to a square matrix that plays an important role in algebraic calculations. It is particularly useful for inverse matrices and solving systems of linear equations.

Inverse Matrix

The inverse of matrix A, A^{-1}, is a matrix such that A·A^{-1} = I, where I is the identity matrix. Inverse exists only for square and non-singular (determinant ≠ 0) matrices.

Applications

Solving systems of linear equations.
In machine learning and neural network computations.
Computer graphics (transformations, rotations).
Cryptography and encryption algorithms.

Practice Exercise

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