Multiplication of Two Determinants

Determinants can also be defined as the scaling factors for matrices. A matric can be defined as an array/ arrangement of numbers in rows and columns. However, there are specific rules for Determinants multiplication. Many mathematical operations are performed on matrices. A determinant is a scalar number associated with every matrix. Many important properties of determinants shall be discussed subsequently in the following topics. There are further uses of determinants as these help in calculating the inverse and adjoint of a matrix that holds utmost importance in complex physical calculations. It is therefore vital to understand the meaning of determinant multiplication.

Define Determinants

A determinant can be defined as the scalar value associated with a square matrix. It is just a number. It helps find out the solutions of linear equations. A determinant can be a real or a complex number for every N x N matrix. The symbol for determinants is where A is any square matrix. Thus, for every matrix A with elements [aij], we can find the Determinant of the matrix. The determinants are helpful in both mathematics and physics calculations as these are tools that make our work easier. The determinant multiplication questions are crucial from an exam point of view, so it is pivotal to understand the rules for its multiplication. 

Determinants multiplication

1. In this section, we will learn how to determine the Determinant for a 2 x 2 and a 3 x 3 square matrix. After that, we shall see how to choose the multiplication of two determinants with determinants multiplication questions. The order of the two determinants has to be the same. 

2. To find the Determinant of a matrix, consider a matrix A with the order of 2 x 2 written as,  

3. The Determinant A can be written as, det A= ad – bc. The solution of ad-bc gives a scalar quantity known as the Determinant of a matrix. 

4. Let us take an example, consider a matrix A1=

The determinant of matrix A1 will be, (2)(3)- (1)(-9) = (6) – (-9) = 6 + 9= 15 which is a scalar number. 

5. Let us now learn how to determine determinants for a 3 x 3 matrix.

Consider a matrix A2 =

The determinant of this 3 x 3 matrix will be,

Det (A2) = ((2) x (10 x 1 – 6 x 1)) – ((5) x (-1x 1 – 3 x 1)) + ((8) x (-1 x 6 – 3 x 10))

Which implies, ((2) x (10-6)) – ((5) x (-1-3)) + ((8)) x (-6-30)

So, (2 x 4) – (5 x -4) + (8 x -36)

Therefore, (8 + 20 – 288) = -260

6. Now, if we have two determinants of second-order expressed as

D1= 

and D2 = 

The Determinants multiplication of D1 and D2 will be done as, 

 Operation Rules

One should keep in mind specific rules while calculating the Determinant’s multiplication. These are as follows,

1. The order of the two determinants has to be the same. 

2. If one wonders what would happen to the value of Determinant if we interchange the rows and columns, then the answer is that there would be no change in the final answer provided the calculations done are correct. 

3. The basic concept is that we follow the row by column rule of Determinants multiplication. Every element of each row of a determinant is multiplied by every element of every column of another determinant. This rule is the same as when two matrices are multiplied.  

4. Since we have already established that the final answer does not change upon interchanging, thus, one can also follow the column by row rule, row by row rule, or column by column rule for multiplication. 

5. The meaning of determinant multiplication holds a property that for two matrix determinants A and B, we can say that AB= AB. 

Conclusion

Here in this article, the reader is now equipped with knowledge about their determinants’ matrices. They can calculate the Determinant of a matrix, and if two determinants are given, the reader can multiply them. Rules of operation are also described in the text. The determinant multiplication questions given in the article help understand the concept more precisely. However, this topic is essential from an exam point of view, so the practice of a lot of questions is necessary since a slight calculation mistake can make the final result incorrect.