Matrix Multiplication Emathhelp (2024)

FAQs

Is matrix multiplication commutative justify your answer? ›

Matrix multiplication is not commutative. It shouldn't be. It corresponds to composition of linear transformations, and composition of func- tions is not commutative. The products aren't the same.

In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. It is faster than the standard matrix multiplication algorithm for large matrices, with a better asymptotic complexity, although the naive algorithm is often better for smaller matrices.

To perform multiplication of two matrices, we should make sure that the number of columns in the 1st matrix is equal to the rows in the 2nd matrix. Therefore, the resulting matrix product will have a number of rows of the 1st matrix and a number of columns of the 2nd matrix.

A trick to help remember matrix multiplication is to rearrange the position of the matrices. It is important to keep in mind that two matrices A and B can be multiplied if and only if the number of columns of A is the same as the number of rows of B.

The product of two matrices will be defined if the number of columns in the first matrix is equal to the number of rows in the second matrix. If the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

Definition(Matrix multiplication)

B = C ||| v 1 v 2 ··· v p ||| D . The product AB is the m × p matrix with columns Av 1 , Av 2 ,..., Av p : AB = C ||| Av 1 Av 2 ··· Av p ||| D . In other words, matrix multiplication is defined column-by-column, or “distributes over the columns of B . ”

Explanation: For matrix multiplication AB , the number of columns of A must be equal to the number of rows of B. A has 3 columns and B has 2 rows, so they are not compatible for multiplication.

The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science. As of January 2024, the best bound on the asymptotic complexity of a matrix multiplication algorithm is O(n^2.371552).

One of the best and easy multiplication tricks for large numbers is to find the tens of one of the numbers, and multiply with that quickly. Adding the remaining leftovers will be easier to calculate fully. E.g., 22 X 83 can be rewritten as (20 X 83) + (2 X 83) which gives us 1660 + 166 = 1826.

Additionally, because the program is run on many data elements and has a high arithmetic intensity, calculations can be used to hide memory access latency in place of big data caches. that's why MATLAB is so fast in matrix Multiplication.

When can you not multiply matrices? ›

Take note that for two matrices to be multiplied the number of columns in the first matrix must equal the number of rows in the second matrix... if this is not so then multiplication is impossible.

Let A = [a_ij] be an m × n matrix and B = [b_jk] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k)^th element c of the matrix C, we take the i^th row of A and k^th column of B, multiply them element-wise and take the sum of all these products.

Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.

One of the best and easy multiplication tricks for large numbers is to find the tens of one of the numbers, and multiply with that quickly. Adding the remaining leftovers will be easier to calculate fully. E.g., 22 X 83 can be rewritten as (20 X 83) + (2 X 83) which gives us 1660 + 166 = 1826.