Matrices are defined as a rectangular array of numbers or functions. Since it is rectangular array, it is 2-dimensional. The two dimensions here are the number of rows (m) and the number of columns (n) respectively. They help define the order of matrix
There are two types of multiplication for matrices: Scalar Multiplication and Matrix Multiplication. Scalar multiplication is easy. You take a regular number called a ‘scalar’ and multiply it with every entry on in the matrix. Let’s see an example:
(This one has two rows and three columns)
Let’s see the matrix multiplication by a single number.
These are the two calculations:
2X4 = 8 & 2X0 = 0
2X1 = 2 & 2x-9 = -18
In the above example, the number 2 is called a scalar, and the process is called ‘Scalar Multiplication.’
As we saw scalar multiplication is easy. Matrix multiplication, however, is another story. You must’ve read complex formulas explaining the process and probably that complex formula didn’t make any sense to you. 🙂 That’s alright. We’ll try and explain matrix multiplication to you in simple terms. So, let us look at some examples of matrix multiplication:
In the above examples, we have matrices A and B. The general notation of a matrix is:
You can see that the matrix is denoted by an upper case letter and its elements are denoted by the same letter in lower case. aij represents any element of the matrix, which is in the ith row and jth column. Similarly, bij represents any element of matrix B. So, a21 represents the element which is in the 2nd row and the 1st column of matrix A or a21 = 21, b32 = 9, b13 = 13 and so on.
Order of Matrix
If a matrix has M rows and N columns, the order of matrix is MxN. We call this an ‘M by N matrix’. The number of rows and columns that a matrix has are called its order or its dimension and by convention, rows are listed first; and columns, second. So, the order of matrix A is 2 × 3 or you can say B is 4 × 3 matrix. If a matrix is of m × n order, it will have mn elements although the converse is not true.
Matrix Multiplication: A direct application for Order of Matrix
The order of matrix finds its direct application in matrix multiplication as it is the key decisive factor in the process. It is only through the order of matrix involved in the calculation that we decide whether the matrices are eligible to get multiplied. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix. Hence, A and B above can’t be multiplied as neither 3=4 (For A x B) or 3=2 (For B x A). This brings another issue:
Matrix multiplication is not commutative
Take into consideration the following example (With different A and B matrices):
The order of matrix here are 2×3 and 3×2. Since the number of columns of A is equal to the number of rows of B, multiplication can take place. If using the above matrices, B had only two rows, its columns would have been too short to multiply against the rows of A. Then AB would not have existed; the product would have been “undefined”. Likewise, if B had, say, four rows, or alternatively if A had two or four columns, then AB would not have existed because A and B would not have been the right sizes.
In other words, assume another matrices P (p1 x q) and Q (q x p2). The multiplication PQ is defined whereas QP is not allowed as p2≠p1.
Matrices have numerous applications in the scientific field as well as practical real life problems. Matrices are used extensively in the study of electrical circuits, quantum mechanics, calculation of battery power outputs, conversion of electrical energy into other useful forms of energy, projection of 3-D image onto a 2-D screen, ranking of pages in Google search, seismic and population related surveys, calculation of an economy’s GDP, and many more.
A study of matrix multiplication is a must for everyone and once you practice them enough, you will be able to do them faster. Happy learning!