Matrix Multiplication: Overview, Questions, Preparation

Multiplication of real numbers is easy but you can also multiply two or more matrices together to form a new matrix. Multiplication of numbers with matrices is also possible but the result that we get from this multiplication is a new matrix with enhanced values. The multiplication of two or more matrices usually results in the formation of a smaller matrix. Matrices can be two-dimensional and one-dimensional. When a matrix is one-dimensional it is a vector and therefore, you can multiply vectors with matrices as well. 

When only one number is multiplied with each element or unit of the matrix then it is called scalar multiplication. However, when you multiply an entire matrix with another matrix then it is known as vector multiplication. 

Definition of Matrix Multiplication 

Two matrices can be multiplied only when the number of rows of one matrix is equal to the number of columns of the second. Let’s say matrix A (aij) is a m x n matrix and matrix B (bjk) is a n x p matrix. Now, the multiplication of these two matrices will result in a new matrix C. 

Therefore, to find the (i, k)th value of the matrix C, we will have to multiply the 1st element of the 1st row of the matrix A i.e. ith row with the 1st element of the 1st column of matrix B i.e. kth row. Similarly, the other elements of the 1st row and other rows of matrix A will be multiplied with the respective columns of the matrix B to find an entirely new matrix C (ckl).

The value that we get by multiplying the elements of the 1st row of matrix A with the elements of matrix B and then adding the obtained results is called a dot product.      

N-tuples & Matrices 

When a matrix is denoted by a list of numbers then it is known as a tuple. Suppose there are n numbers in a matrix then it will be denoted as an n-tuple. 

1. In the matrix A = [2 5], you can think of each row and column as a tuple. 
                                [5 1]

Solution.

Therefore, row1, row2, column1, and column2 are 4 tuples. Row1 will be referred to as r1 = (2, 5) and row2 will be referred to as r2 = (5, 1). Similarly, column1 is denoted by c1 = (2, 5) and column2 is denoted by c2 = (5, 1).
(3.6)

2. Let's say that A = [1 4 6] and B = [7]
                                                            [8]
                                                            [9]

Solution.

Therefore, the dot product will be C = [1x7 + 4x8 + 6x9] = [7 + 32 + 54] = [93]
(3.8)

3. When we multiply a real number with a matrix, it is known as scalar multiplication. The example of scalar multiplication is given below:

 5 x [5 2] = [5 x 5  2 x 5] = [25  10]
       [4 5]    [5 x 4  5 x 5]     [20  25]    

Q: When can you not multiply one matrix with another matrix?

Q: Is it possible to take the dot product of n-tuples of various lengths?

Q: If X and Y are two matrices then is X.Y = Y.X?

Q: What can multiplication of matrices be used for?

Q: How to multiply square matrices?