Basic Matrix Operations

Basic Matrix Operations A matrix is a rectangular arrangement of numbers organized in rows and columns. Matrices are widely used in mathematics, physics, engineering, and computer science. A matrix is usually denoted by a capital letter such as A or B. The size of a matrix is described by its number of rows and columns (m × n). Example: A = [[1, 2], [3, 4]] is a 2×2 matrix. Matrices are used to solve systems of equations, perform transformations, and represent data. Matrix Addition Two matrices can be added only if they have the same dimensions. Rule: Add corresponding elements. Example: A = [[1, 2], [3, 4]] B = [[5, 6], [7, 8]] A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]] Matrix Subtraction Matrix subtraction is similar to addition. Rule: Subtract corresponding elements. Example: A = [[8, 7], [6, 5]] B = [[1, 2], [3, 4]] A - B = [[8-1, 7-2], [6-3, 5-4]] = [[7, 5], [3, 1]] Scalar Multiplication A scalar is a single number. When multiplying a matrix by a scalar, multiply each element. Example: k = 2, A = [[1, 2], [3, 4]] 2A = [[2×1, 2×2], [2×3, 2×4]] = [[2, 4], [6, 8]] Matrix Multiplication Matrix multiplication is possible when the number of columns of the first matrix equals the number of rows of the second matrix. Example: A = [[1, 2], [3, 4]] B = [[2, 0], [1, 2]] AB = [[(1×2 + 2×1), (1×0 + 2×2)], [(3×2 + 4×1), (3×0 + 4×2)]] AB = [[4, 4], [10, 8]] Transpose of a Matrix The transpose of a matrix is obtained by switching rows with columns. Example: A = [[1, 2, 3], [4, 5, 6]] A^T = [[1, 4], [2, 5], [3, 6]] This operation is useful in many applications like linear algebra and machine learning.