A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are fundamental to linear algebra and are used to represent linear transformations, systems of linear equations, and data in various scientific and engineering applications. The size of a matrix is defined by its number of rows and columns (e.g., a 2×3 matrix has 2 rows and 3 columns).

To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B. The element in the i-th row and j-th column of the result matrix C is calculated by taking the dot product of the i-th row of A and the j-th column of B: C[i,j] = Σ A[i,k] × B[k,j]. Unlike regular numbers, matrix multiplication is generally not commutative (A × B ≠ B × A).

The determinant is a scalar value that can be computed from the elements of a square matrix. It encodes certain properties of the linear transformation described by the matrix. For a 2×2 matrix [[a, b], [c, d]], the determinant is ad – bc. A matrix is invertible if and only if its determinant is non-zero. The determinant is also used to find the area/volume scaling factor of transformations and to solve systems of equations using Cramer’s rule.

For a 2×2 matrix A = [[a, b], [c, d]], the inverse A⁻¹ is found using the formula: A⁻¹ = (1/det(A)) × [[d, -b], [-c, a]]. First, calculate the determinant (ad – bc). If the determinant is zero, the matrix has no inverse (it is singular). If non-zero, swap the elements on the main diagonal (a and d), change the signs of the off-diagonal elements (b and c), and multiply the entire matrix by 1/determinant.

No, matrix addition is only defined for matrices of the exact same dimensions. To add two matrices, they must have the same number of rows and the same number of columns. The sum is obtained by adding the corresponding elements from each matrix: C[i,j] = A[i,j] + B[i,j]. If the dimensions differ, the operation is undefined.

The transpose of a matrix is formed by turning all the rows of the matrix into columns and vice versa. If A is an m×n matrix, its transpose Aᵀ is an n×m matrix where Aᵀ[i,j] = A[j,i]. For example, the transpose of a 2×3 matrix is a 3×2 matrix. The transpose operation is widely used in linear algebra, statistics (e.g., covariance matrices), and solving least squares problems.

An identity matrix (denoted as I or Iₙ) is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts as the multiplicative identity for matrices, meaning that for any square matrix A of the same size, A × I = I × A = A. For example, the 2×2 identity matrix is [[1, 0], [0, 1]]. It is analogous to the number 1 in standard arithmetic multiplication.

Matrices are used extensively in computer graphics to perform 3D transformations (rotation, scaling, translation), in physics and engineering to model systems and circuits, in economics for input-output models, in machine learning and data science for handling datasets and neural network weights, and in cryptography for encrypting and decrypting messages. They are a foundational tool for organizing and manipulating multidimensional data.