Multiply two matrices

Learning Objectives Determine if two matrices can be multiplied by checking their dimensions. State the dimensions of the resulting product matrix. Calculate the dot product of a row vector and a column vector. Accurately multiply two 2x2 matrices. Multiply matrices of other compatible dimensions (e.g., 2x3 and 3x2). Recognize and demonstrate that matrix multiplication is not commutative (i.e., AB ≠ BA). Ever wonder how game developers make a character jump and rotate in 3D space, or how businesses track sales across multiple stores? 🎮 It's all done with the power of multiplying matrices! In this tutorial, you'll learn the specific step-by-step process for multiplying two matrices. This operation is different from regular multiplication, but it's a fundament...

TermDefinitionExample MatrixA rectangular grid of numbers, symbols, or expressions, arranged in rows and columns.Matrix A = [[5, 8], [2, 0]] Dimensions of a MatrixThe size of a matrix, described by its number of rows and columns, written as 'rows × columns'.The matrix [[1, 2, 3], [4, 5, 6]] has dimensions 2 × 3 because it has 2 rows and 3 columns. ElementA single number or entry within a matrix. An element is identified by its row and column position, denoted as aᵢⱼ (row i, column j).In matrix A = [[5, 8], [2, 0]], the element a₂₁ is 2. Dot ProductThe result of multiplying the corresponding entries of a row from the first matrix and a column from the second matrix, and then summing those products.The dot product of row [1, 2] and column [[3], [4]] is (1 * 3) + (2 * 4) = 3 + 8 =...

The Compatibility Rule To multiply matrix A by matrix B (to find AB), the number of columns in A must be equal to the number of rows in B. This is the first and most important check. If A has dimensions m × n, B must have dimensions n × p. If the 'inner' numbers (n and n) don't match, the matrices cannot be multiplied. Dimensions of the Product Matrix If A is an m × n matrix and B is an n × p matrix, the product matrix AB will have dimensions m × p. The dimensions of the resulting matrix are determined by the 'outer' numbers of the original matrices' dimensions. This tells you the size of your answer matrix before you even start calculating. The Multiplication Formula If C = AB, the element in the i-th row and j-th column of C (cᵢⱼ) is the...