Matrix multiplication

Matrix Multiplication

  • Matrix multiplication is an operation in mathematics where two matrices are multiplied to give a third matrix.
  • It’s not commutative; the order in which matrices are multiplied can change the result. In other words, AB ≠ BA.
  • The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.

Rules for Multiplying Matrices

  • The element in the i-th row and j-th column of the resulting matrix is given by multiplying the i-th row elements of the first matrix by the j-th column elements of the second matrix and summing them up.
  • If matrix A is of size (m x n) and matrix B is of size (n x p), then the resultant matrix C = A x B will be of size (m x p).

Matrix Multiplication Techniques

  • To multiply two matrices, take the first row of the first matrix and the first column of the second, multiply corresponding elements and sum up all those products. This sum will be the element at the first row and first column of the result matrix.
  • Continue this process with the remaining rows of the first matrix and columns of the second matrix until every row of the first matrix has been multiplied with every column of the second matrix.
  • If there’s a row in the first matrix that has as many entries as there are entries in a column of the second matrix, but the sum of the products is nonzero, the entry in the new matrix is the sum of those products.

Identity Matrix and Multiplicative Identity

  • The Identity Matrix is a square matrix in which all the elements of the principal (main) diagonal are ones and all other elements are zeros.
  • When a matrix is multiplied by an Identity matrix, the result is the original matrix. Hence it’s also called the Multiplicative Identity in matrix algebra.

Zero Matrix and Multiplicative Zero

  • A matrix where all its elements are zero is called a Zero Matrix.
  • When the Zero Matrix multiplies any matrix, the product is always the Zero Matrix. Hence, it’s often referred as the Multiplicative Zero.

Applying Matrix Multiplication in Systems of Linear Equations

  • Matrix multiplication can be used to solve systems of linear equations by converting them to matrix form. This technique is known in mathematics as Matrix Algebra.