Matrix Multiplication How To Find The Product Of AB And BA

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Matrix multiplication is a fundamental operation in linear algebra, with applications spanning various fields, including computer graphics, data analysis, and physics. Understanding how to multiply matrices is crucial for solving systems of linear equations, performing transformations, and analyzing complex data sets. In this article, we will delve into the process of matrix multiplication, focusing on calculating the product of two matrices, ABAB and BABA, where AA and BB are given matrices. We will explore the conditions for matrix multiplication to be defined, the steps involved in the calculation, and the significance of the order of multiplication. Furthermore, we will provide detailed examples to illustrate the concepts and techniques involved in matrix multiplication. This comprehensive guide aims to equip you with the knowledge and skills necessary to confidently perform matrix multiplication and apply it to real-world problems.

In matrix algebra, the order in which matrices are multiplied matters significantly. Unlike scalar multiplication, where ab=baab = ba, matrix multiplication is not commutative in general, meaning that ABAB is not necessarily equal to BABA. This distinction arises from the fact that matrix multiplication involves a row-by-column operation, where the elements of the rows of the first matrix are multiplied by the corresponding elements of the columns of the second matrix, and the products are summed. The dimensions of the matrices involved play a critical role in determining whether the multiplication is defined and what the dimensions of the resulting matrix will be. Specifically, for the product ABAB to be defined, the number of columns in matrix AA must be equal to the number of rows in matrix BB. The resulting matrix will then have the same number of rows as AA and the same number of columns as BB. The product BABA can only be found if the number of columns in B is equal to the number of rows in A.

Let's consider two matrices, AA and BB. If AA is an mimesnm imes n matrix (meaning it has mm rows and nn columns) and BB is a pimesqp imes q matrix, then the product ABAB is defined only if n=pn = p. In this case, the resulting matrix ABAB will be of size mimesqm imes q. Similarly, the product BABA is defined only if q=mq = m, and the resulting matrix BABA will be of size pimesnp imes n. The elements of the product matrix are calculated by taking the dot product of the rows of the first matrix and the columns of the second matrix. This involves multiplying corresponding elements and then summing the results. The order of multiplication is crucial because it affects the dimensions of the resulting matrix and the specific elements that are multiplied. If ABAB is defined, BABA may not be defined, and even if both ABAB and BABA are defined, they may not have the same dimensions, and their elements may be different.

a. Finding the Product of AB and BA

Given the matrices A=(240)A=\begin{pmatrix}2 & 4 & 0\end{pmatrix} and B=[234]B=\begin{bmatrix}2 \\ 3 \\ 4\end{bmatrix}, we aim to find the products ABAB and BABA. Matrix AA is a 1imes31 imes 3 matrix (1 row and 3 columns), while matrix BB is a 3imes13 imes 1 matrix (3 rows and 1 column). To compute the product ABAB, we multiply the rows of AA by the columns of BB. Since the number of columns in AA (3) is equal to the number of rows in BB (3), the product ABAB is defined, and the resulting matrix will be of size 1imes11 imes 1. To calculate ABAB, we perform the following multiplication and summation: (2imes2)+(4imes3)+(0imes4)=4+12+0=16(2 imes 2) + (4 imes 3) + (0 imes 4) = 4 + 12 + 0 = 16. Therefore, AB=(16)AB = \begin{pmatrix}16\end{pmatrix}, which is a 1imes11 imes 1 matrix.

Now, let's consider the product BABA. To compute BABA, we multiply the rows of BB by the columns of AA. Since the number of columns in BB (1) is not equal to the number of rows in AA (1), the product BABA is defined, and the resulting matrix will be of size 3imes33 imes 3. To calculate BABA, we perform the following multiplications: The first row of BB multiplied by the first column of AA gives us the element in the first row and first column of BABA: 2imes2=42 imes 2 = 4. The first row of BB multiplied by the second column of AA gives us the element in the first row and second column of BABA: 2imes4=82 imes 4 = 8. The first row of BB multiplied by the third column of AA gives us the element in the first row and third column of BABA: 2imes0=02 imes 0 = 0. The second row of BB multiplied by the first column of AA gives us the element in the second row and first column of BABA: 3imes2=63 imes 2 = 6. The second row of BB multiplied by the second column of AA gives us the element in the second row and second column of BABA: 3imes4=123 imes 4 = 12. The second row of BB multiplied by the third column of AA gives us the element in the second row and third column of BABA: 3imes0=03 imes 0 = 0. The third row of BB multiplied by the first column of AA gives us the element in the third row and first column of BABA: 4imes2=84 imes 2 = 8. The third row of BB multiplied by the second column of AA gives us the element in the third row and second column of BABA: 4imes4=164 imes 4 = 16. The third row of BB multiplied by the third column of AA gives us the element in the third row and third column of BABA: 4imes0=04 imes 0 = 0. Therefore, BA=[48061208160]BA = \begin{bmatrix}4 & 8 & 0 \\ 6 & 12 & 0 \\ 8 & 16 & 0\end{bmatrix}, which is a 3imes33 imes 3 matrix. This example illustrates that even though both ABAB and BABA are defined, they result in matrices of different sizes and different elements, highlighting the non-commutative nature of matrix multiplication.

b. Calculating AB and BA for Another Pair of Matrices

Now, let's consider another example with matrices A=[1βˆ’23βˆ’425]A=\begin{bmatrix}1 & -2 & 3 \\ -4 & 2 & 5\end{bmatrix} and B=[23βˆ’104βˆ’2]B=\begin{bmatrix}2 & 3 \\ -1 & 0 \\ 4 & -2\end{bmatrix}. Matrix AA is a 2imes32 imes 3 matrix, and matrix BB is a 3imes23 imes 2 matrix. To find the product ABAB, we multiply the rows of AA by the columns of BB. Since the number of columns in AA (3) is equal to the number of rows in BB (3), the product ABAB is defined, and the resulting matrix will be of size 2imes22 imes 2. To calculate ABAB, we perform the following multiplications and summations: The first row of AA multiplied by the first column of BB gives us the element in the first row and first column of ABAB: (1imes2)+(βˆ’2imesβˆ’1)+(3imes4)=2+2+12=16(1 imes 2) + (-2 imes -1) + (3 imes 4) = 2 + 2 + 12 = 16. The first row of AA multiplied by the second column of BB gives us the element in the first row and second column of ABAB: (1imes3)+(βˆ’2imes0)+(3imesβˆ’2)=3+0βˆ’6=βˆ’3(1 imes 3) + (-2 imes 0) + (3 imes -2) = 3 + 0 - 6 = -3. The second row of AA multiplied by the first column of BB gives us the element in the second row and first column of ABAB: (βˆ’4imes2)+(2imesβˆ’1)+(5imes4)=βˆ’8βˆ’2+20=10(-4 imes 2) + (2 imes -1) + (5 imes 4) = -8 - 2 + 20 = 10. The second row of AA multiplied by the second column of BB gives us the element in the second row and second column of ABAB: (βˆ’4imes3)+(2imes0)+(5imesβˆ’2)=βˆ’12+0βˆ’10=βˆ’22(-4 imes 3) + (2 imes 0) + (5 imes -2) = -12 + 0 - 10 = -22. Therefore, AB=[16βˆ’310βˆ’22]AB = \begin{bmatrix}16 & -3 \\ 10 & -22\end{bmatrix}, which is a 2imes22 imes 2 matrix.

Next, we compute the product BABA. To find BABA, we multiply the rows of BB by the columns of AA. Since the number of columns in BB (2) is not equal to the number of rows in AA (2), the product BABA is defined, and the resulting matrix will be of size 3imes33 imes 3. To calculate BABA, we perform the following multiplications: The first row of BB multiplied by the first column of AA gives us the element in the first row and first column of BABA: (2imes1)+(3imesβˆ’4)=2βˆ’12=βˆ’10(2 imes 1) + (3 imes -4) = 2 - 12 = -10. The first row of BB multiplied by the second column of AA gives us the element in the first row and second column of BABA: (2imesβˆ’2)+(3imes2)=βˆ’4+6=2(2 imes -2) + (3 imes 2) = -4 + 6 = 2. The first row of BB multiplied by the third column of AA gives us the element in the first row and third column of BABA: (2imes3)+(3imes5)=6+15=21(2 imes 3) + (3 imes 5) = 6 + 15 = 21. The second row of BB multiplied by the first column of AA gives us the element in the second row and first column of BABA: (βˆ’1imes1)+(0imesβˆ’4)=βˆ’1+0=βˆ’1(-1 imes 1) + (0 imes -4) = -1 + 0 = -1. The second row of BB multiplied by the second column of AA gives us the element in the second row and second column of BABA: (βˆ’1imesβˆ’2)+(0imes2)=2+0=2(-1 imes -2) + (0 imes 2) = 2 + 0 = 2. The second row of BB multiplied by the third column of AA gives us the element in the second row and third column of BABA: (βˆ’1imes3)+(0imes5)=βˆ’3+0=βˆ’3(-1 imes 3) + (0 imes 5) = -3 + 0 = -3. The third row of BB multiplied by the first column of AA gives us the element in the third row and first column of BABA: (4imes1)+(βˆ’2imesβˆ’4)=4+8=12(4 imes 1) + (-2 imes -4) = 4 + 8 = 12. The third row of BB multiplied by the second column of AA gives us the element in the third row and second column of BABA: (4imesβˆ’2)+(βˆ’2imes2)=βˆ’8βˆ’4=βˆ’12(4 imes -2) + (-2 imes 2) = -8 - 4 = -12. The third row of BB multiplied by the third column of AA gives us the element in the third row and third column of BABA: (4imes3)+(βˆ’2imes5)=12βˆ’10=2(4 imes 3) + (-2 imes 5) = 12 - 10 = 2. Therefore, BA=[βˆ’10221βˆ’12βˆ’312βˆ’122]BA = \begin{bmatrix}-10 & 2 & 21 \\ -1 & 2 & -3 \\ 12 & -12 & 2\end{bmatrix}, which is a 3imes33 imes 3 matrix. This example further demonstrates the non-commutative property of matrix multiplication, as ABAB and BABA have different dimensions and elements.

Conclusion: Key Takeaways from Matrix Multiplication

In summary, matrix multiplication is a fundamental operation in linear algebra with significant applications in various fields. The key to understanding matrix multiplication lies in recognizing the importance of the order of multiplication and the dimensions of the matrices involved. The product ABAB is defined only if the number of columns in matrix AA is equal to the number of rows in matrix BB, and the resulting matrix has the same number of rows as AA and the same number of columns as BB. Similarly, the product BABA is defined only if the number of columns in BB is equal to the number of rows in AA. The elements of the product matrix are calculated by taking the dot product of the rows of the first matrix and the columns of the second matrix.

One of the most crucial aspects of matrix multiplication is its non-commutative nature. In general, ABAB is not equal to BABA. This is because matrix multiplication involves a row-by-column operation, and changing the order of multiplication changes the way the elements are combined. In some cases, ABAB may be defined, while BABA is not, or vice versa. Even when both ABAB and BABA are defined, they may have different dimensions or different elements. This non-commutative property is a key distinction between matrix multiplication and scalar multiplication, where the order of multiplication does not affect the result.

Through the examples provided, we have illustrated the step-by-step process of calculating the products ABAB and BABA for different pairs of matrices. These examples highlight the importance of carefully checking the dimensions of the matrices before attempting to multiply them. They also demonstrate how the elements of the product matrix are determined by the dot products of the rows and columns of the original matrices. Understanding these concepts and techniques is essential for mastering matrix multiplication and applying it to various problems in mathematics, science, and engineering.

In conclusion, the ability to confidently perform matrix multiplication is a valuable skill in many areas. By understanding the conditions for matrix multiplication, the steps involved in the calculation, and the non-commutative nature of the operation, you can effectively utilize matrices to solve complex problems and gain insights from data. Whether you are working with systems of equations, transformations, or data analysis, matrix multiplication is a powerful tool that can help you achieve your goals.