Problem 32 - College Algebra

Question

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 5 & 0 & -2 \\ 3 & -2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & 1 & 0 \\ 1 & -4 & 5 \\ 3 & -1 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

This gives us the matrices AB and BA as \[\begin{bmatrix} 2 & 5 & -3 \ -1 & 7 & -4 \ 3 & 11 & -6 \end{bmatrix}\] and \[\begin{bmatrix} 6 & -1 & 0 \ 8 & -7 & 14 \ 4 & -1 & 10 \end{bmatrix}\] respectively.

Step 1: Calculate AB

To calculate AB, multiply each row in A with each column in B and add up the results. This yields: \[\begin{bmatrix} (1*1)+ ((-1)*1)+ (1*3) & (1*1)+ ((-1)*-4)+ (1*-1) & (1*0)+ ((-1)*5)+ (1*2) \ (5*1)+ (0*1)+ ((-2)*3) & (5*1)+ (0*-4)+ ((-2)*-1) & (5*0)+ (0*5)+ ((-2)*2) \ (3*1)+ ((-2)*1)+ (2*3) & (3*1)+ ((-2)*-4)+ (2*-1) & (3*0)+ ((-2)*5)+ (2*2) \end{bmatrix}\]

Step 2: Simplify matrix AB

By doing the calculations, we eventually find that AB equals: \[\begin{bmatrix} 2 & 5 & -3 \ -1 & 7 & -4 \ 3 & 11 & -6 \end{bmatrix}\]

Step 3: Calculate BA

Following the same process but swapping the order of A and B, BA is: \[\begin{bmatrix} (1*1)+(1*5)+(0*3) & ((-1)*1)+(5*0)+(0*-2) & (1*1)+(5*-2)+(0*2) \ (1*1)+((-4)*5)+(5*3) & ((-1)*1)+(0*5)+(5*-2) & (1*1)+((-4)*-2)+(5*2) \ (3*1)+((-1)*5)+(2*3) & (3*-1)+(1*0)+((-1)*-2) & (3*1)+((-1)*-2)+(2*2) \end{bmatrix}\]

Step 4: Simplify matrix BA

After performing the calculations, we find that BA equal to: \[\begin{bmatrix} 6 & -1 & 0 \ 8 & -7 & 14 \ 4 & -1 & 10 \end{bmatrix}\]

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. In high school and college algebra, we learn to solve equations and understand the properties of and operations with numbers and expressions. Specifically, matrix multiplication falls under the umbrella of linear algebra, which is a significant area within college-level algebra. Here, we delve into operations dealing with matrices, representing systems of linear equations. Understanding the rules governing matrix operations is crucial, as they differ markedly from those for ordinary arithmetic.

For instance, while multiplying two matrices, one must align rows of the first matrix with columns of the second matrix, a process that does not have an analogue in simple number multiplication. This procedure helps in solving complex problems in various fields including engineering, physics, and computer science.

Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements. Matrices are not just a mathematical abstraction, but they serve as powerful tools for representing and solving real-world problems. In our context, matrices are used to linearly transform data, solve systems of equations, and represent graphs and networks.

Understanding a matrix's dimensions, given by the number of rows followed by the number of columns, is key to executing operations like matrix multiplication correctly. The capability to perform matrix operations accurately is a fundamental skill that students in algebra and beyond must develop to prepare for more advanced subjects in mathematics and applied sciences.

College Algebra

College algebra extends beyond the algebra taught at the high school level. It covers a wider array of topics and delves deeper into the concepts of function theory, complex numbers, and matrices. In college algebra, matrix multiplication is an essential concept that finds applications in various disciplines. It is crucial not only to understand how to perform the multiplication step by step but also to appreciate the implications and the limitations of such operations.

For example, matrix multiplication is not commutative, which means that the order in which matrices are multiplied matters—a fundamental concept highlighted in the exercise. It's also worth noting the conditions required for matrix multiplication to be defined: the number of columns in the first matrix must match the number of rows in the second matrix. Insight into these high-level algebra concepts empowers students to tackle real-world problems involving linear transformations and vector spaces.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and finding the inverse of matrices. Among these, matrix multiplication is a particularly vital operation and it can seem daunting due to its complexity compared to scalar multiplication. The exercise involves calculating the product of two square matrices, which is a fundamental matrix operation. The step-by-step solution demonstrates the process of multiplying each row of the first matrix (A) by each column of the second matrix (B), summing the products to obtain individual elements of the resulting matrix.

It's essential to understand that the associative and distributive properties hold for matrix multiplication, but, as previously mentioned, the commutative property does not. Additionally, not every matrix has an inverse, and only square matrices can be multiplied by themselves—a concept that branches off into the topic of matrix powers and exponentiation. Familiarity with these concepts is foundational for anyone studying linear algebra, as these operations are integral to understanding how matrices can represent and solve systems of linear equations.