To multiply matrices {matrix multiplication} {matrix dot product}, multiply each row by each column. Matrix with m columns and n rows times matrix with n columns and p rows makes matrix of m columns and p rows. First-matrix rows and second-matrix columns must have same rank. For 1x1 matrices [a11] and [b11], matrix dot product is [a11*b11]. For 2x2 matrices [a11 a12 / a21 a22] and [b11 b12 / b21 b22], matrix dot product is [a11*b11 + a11*b21 a12*b12 + a12*b22 / a21*b11 + a21*b21 a22*b12 + a22*b22]. For example, [1 2 / 3 4] . [5 4 / 3 2] = [1*5+1*3 2*4+2*2 / 3*5+3*3 4*4+4*2].
vector
Vector dot products are matrix multiplications of one-row 1xN matrix with one-column Nx1 matrix.
properties
Matrix multiplication is not commutative but is associative.
purposes
Multiplying matrices indicates results of interactions between two effects. Squaring matrix is like repeating operation.
Mathematical Sciences>Algebra>Equation>System>Matrix>Operations
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Date Modified: 2022.0224