Problem 47 - Precalculus with Limits

Question

In Exercises 47-52, if possible, find (a) \(AB\), (b) \(BA\), and (c) \(A^2\). (Note: \(A^2 = AA\).) \(A=\left[\begin{array}{r} 1 && 2 \\ 4 && 2 \end{array}\right]\), \(B=\left[\begin{array}{r} 2 & -1 \\ -1 & 8 \end{array}\right]\)

Step-by-Step Solution

Answer

\(AB = \left[\begin{array}{rr} 0 && 15 \ 6 && 12 \end{array}\right]\), \(BA = \left[\begin{array}{rr} -2 && 2 \ 31 && 12 \end{array}\right]\) and \(A^2 = \left[\begin{array}{rr} 9 && 6 \ 12 && 8 \end{array}\right]\)

Step 1: Calculate AB

To find the product of two matrices, each element of the first row of the first matrix multiplies with each element of the first column of the second matrix and then added together to get the first element for the resultant matrix. So, \(AB = \left[\begin{array}{rr} 1*2+2*-1 && 1*-1+2*8 \ 4*2+2*-1 && 4*-1+2*8 \end{array}\right] = \left[\begin{array}{rr} 0 && 15 \ 6 && 12 \end{array}\right]\)

Step 2: Calculate BA

The procedure is the same as Step 1, but here the second matrix multiplies with the first matrix. \(BA = \left[\begin{array}{rr} 2*1+(-1)*4 && 2*2+(-1)*2 \ -1*1+4*8 && -1*2+8*2 \end{array}\right] = \left[\begin{array}{rr} -2 && 2 \ 31 && 12 \end{array}\right]\)

Step 3: Calculate \(A^2\)

Calculate this by multiplying the matrix A by itself. \(A^2 = AA = \left[\begin{array}{rr} 1*1+2*4 && 1*2+2*2 \ 4*1+2*4 && 4*2+2*2 \end{array}\right] = \left[\begin{array}{rr} 9 && 6 \ 12 && 8 \end{array}\right]\)

Key Concepts

Matrices Operations

Matrix multiplication is a fundamental operation in linear algebra, crucial for solving a wide range of mathematical problems, especially in the fields of engineering, physics, and computer science. Unlike scalar multiplication, which involves multiplying each element of a matrix by a single number, matrix multiplication combines two matrices, named 'A' and 'B', into a new matrix 'C'.

To perform the multiplication of matrices 'A' and 'B', we have to follow specific rules. Each element of the resulting matrix 'C' is computed by taking the dot product of the corresponding row from matrix 'A' with the corresponding column from matrix 'B'. It's essential to know that for matrices to be multipliable, the number of columns in the first matrix must match the number of rows in the second.

• For the product AB, we multiply row elements of A by the corresponding column elements of B and sum them up to get each element of the resulting matrix.
• For the product BA, we reverse the roles of A and B.
• The product A², also indicated as AA, means multiplying matrix A by itself, which is only possible if A is square (same number of rows and columns).

Remember, the order of multiplication matters, AB is generally not equal to BA, as demonstrated by their different resulting matrices in our problem.

Precalculus

Precalculus serves as the bridge between Algebra and Calculus, providing students with the necessary foundation in functions, sequences, and mathematical analysis. It deepens understanding of algebraic structures and introduces the concept of limits, which is pivotal in Calculus. In the context of matrices, precalculus introduces the idea of matrix operations as a means to solve systems of equations and transformations in the plane.

Understanding how to manipulate algebraic expressions is crucial in performing matrix operations. Proficiency in precalculus ensures that a student can navigate through the intricacies of matrix multiplication, such as knowing when two matrices can be multiplied and the differences in their products, AB versus BA, and their interpretations in different mathematical contexts. It lays the groundwork for later coursework in linear algebra and multivariable calculus where matrices become indispensable tools.

Algebraic Structures

Algebraic structures come in various forms such as groups, rings, and fields, providing a framework for different mathematical entities and their operations. Matrices fall under the umbrella of these structures as they build on the principles of vector spaces and linear transformations, both of which are fundamental concepts in abstract algebra.

A matrix itself is an array of numbers, but it represents much more than that when examined through the lens of different algebraic structures. It can be seen as an operator on vector spaces, transforming vectors from one space to another. The rules of matrix multiplication, where the consistency of dimensionality is key, reflect the rigorous formalism found in algebra. Moreover, the non-commutative nature of matrix multiplication (where AB ≠ BA) is an example of an algebraic structure where the commutative property does not always apply.
Ultimately, exploring matrices within the framework of algebraic structures broadens one's appreciation for the depth and breadth of mathematics as it relates to numerous real-world applications.