Problem 12 - Algebra 2

Question

Use \(A=\left[\begin{array}{cc}{2} & {-1} \\ {3} & {5}\end{array}\right], B=\left[\begin{array}{cc}{-4} & {1} \\ {8} & {0}\end{array}\right]\) and \(C=\left[\begin{array}{cc}{3} & {2} \\ {-1} & {2}\end{array}\right]\) to determine whether the following equations are true for the given matrices. \(A B=B A\)

Step-by-Step Solution

Answer

The equation \(AB = BA\) is false because \(AB \neq BA\).

Step 1: Calculate AB

To determine if the equation \(AB = BA\) is true, we first need to compute the matrix multiplication of \(A\) and \(B\).The matrix \(A\) is \(\begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix}\) and matrix \(B\) is \(\begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix}\).The product \(AB\) is calculated as follows:\[AB = \begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix} \begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix} = \begin{bmatrix} (2)(-4) + (-1)(8) & (2)(1) + (-1)(0) \ (3)(-4) + (5)(8) & (3)(1) + (5)(0) \end{bmatrix}\]\[AB = \begin{bmatrix} -8 - 8 & 2 + 0 \ -12 + 40 & 3 + 0 \end{bmatrix} = \begin{bmatrix} -16 & 2 \ 28 & 3 \end{bmatrix}\]

Step 2: Calculate BA

Next, calculate the matrix multiplication of \(B\) and \(A\) to compare it with \(AB\).Multiply \(B = \begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix}\) by \(A = \begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix}\):\[BA = \begin{bmatrix} -4 & 1 \ 8 & 0 \end{bmatrix} \begin{bmatrix} 2 & -1 \ 3 & 5 \end{bmatrix} = \begin{bmatrix} (-4)(2) + (1)(3) & (-4)(-1) + (1)(5) \ (8)(2) + (0)(3) & (8)(-1) + (0)(5) \end{bmatrix}\]\[BA = \begin{bmatrix} -8 + 3 & 4 + 5 \ 16 + 0 & -8 + 0 \end{bmatrix} = \begin{bmatrix} -5 & 9 \ 16 & -8 \end{bmatrix}\]

Step 3: Compare AB and BA

Now, compare the results of \(AB\) and \(BA\):\[AB = \begin{bmatrix} -16 & 2 \ 28 & 3 \end{bmatrix}\] and \[BA = \begin{bmatrix} -5 & 9 \ 16 & -8 \end{bmatrix}\].\(AB\) and \(BA\) are not equal, as their corresponding elements differ. For example, the top left elements are -16 and -5, respectively.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in algebra, especially when it comes to operations like addition and multiplication. It suggests that the order in which we perform the operation does not affect the result. For example, in basic arithmetic, we know that

• Addition: \(a + b = b + a\)
• Multiplication: \(a \times b = b \times a\)

However, this property does not universally apply to all mathematical operations. Matrix multiplication is a notable exception. Unlike numbers, matrices do not generally follow the commutative property during multiplication. This means that for matrices \(A\) and \(B\), \(AB\) might not equal \(BA\). This was clearly shown in our exercise where the products \(AB = \begin{bmatrix} -16 & 2 \ 28 & 3 \end{bmatrix}\) and \(BA = \begin{bmatrix} -5 & 9 \ 16 & -8 \end{bmatrix}\) were not equal, demonstrating that matrix multiplication is not commutative.

Matrices

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are useful for organizing information and for performing a variety of algebraic operations. They can be of different dimensions, such as 2x2, 3x3, or any other form defined by the number of rows and columns. Each element of a matrix is accessed and identified by its position in the rows and columns.
Matrices are extensively used in various fields.

• Engineering and computer graphics for transformations and rotations.
• Statistics to represent and manage datasets.
• Physics for quantum mechanics and systems modeling.

Although matrices can hold any number of data entries depending on their size, a significant tool we use is matrix multiplication. In this operation, one must understand how to appropriately align and compute products between matrices. Unlike simple numbers, the order of multiplying matrices matters. This is why, in our exercise, calculating \(AB\) and \(BA\) gave different results.

Algebraic Expressions

Algebraic expressions play a crucial role in understanding and solving mathematical problems, particularly with matrices. An algebraic expression is a combination of constants, variables, and operations. In matrix algebra, algebraic expressions involve various operations such as addition, subtraction, and multiplication. It's important to differentiate these expressions from the traditional numeric algebra as they introduce new layers of complexity.

• Order and dimensions must be consistent for operations to take place.
• Matrices must be compatible in terms of their dimensions to perform multiplication (i.e., the number of columns in the first matrix must equal the number of rows in the second matrix).

For instance, when working through the given problem, we observed the process of matrix multiplication, which is more complex than multiplying two numbers. Algebraic expressions guide us through the rules and methods required to achieve correct solutions in matrix operations, much like they do in other algebraic contexts.