Problem 31 - Algebra for College Students

Question

For Problems \(31-34\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} 2 & 4 \\ 5 & -3 \end{array}\right] \quad B=\left[\begin{array}{ll} -2 & 3 \\ -1 & 2 \end{array}\right] \\ &C=\left[\begin{array}{ll} 2 & 1 \\ 3 & 7 \end{array}\right] \end{aligned}\) $$ \text { Show that }(A B) C=A(B C) \text {. } $$

Step-by-Step Solution

Answer

The associative property is verified: \( (AB)C = A(BC) \).

Step 1: Define the Matrices

First, write down the matrices given in the problem: \( A = \begin{bmatrix} 2 & 4 \ 5 & -3 \end{bmatrix} \), \( B = \begin{bmatrix} -2 & 3 \ -1 & 2 \end{bmatrix} \), and \( C = \begin{bmatrix} 2 & 1 \ 3 & 7 \end{bmatrix} \).

Step 2: Multiply A and B

To find \( AB \), multiply matrix \( A \) by matrix \( B \). The element in the first row and first column is calculated as \( 2(-2) + 4(-1) = -4 - 4 = -8 \). For the first row and second column, it's \( 2(3) + 4(2) = 6 + 8 = 14 \). For the second row and first column, it's \( 5(-2) + (-3)(-1) = -10 + 3 = -7 \). For the second row and second column, it's \( 5(3) + (-3)(2) = 15 - 6 = 9 \). Thus, \( AB = \begin{bmatrix} -8 & 14 \ -7 & 9 \end{bmatrix} \).

Step 3: Multiply (AB) with C

Multiply \( (AB) \) with \( C \). The first element is \( -8(2) + 14(3) = -16 + 42 = 26 \), second element is \( -8(1) + 14(7) = -8 + 98 = 90 \). Third element is \( -7(2) + 9(3) = -14 + 27 = 13 \), fourth element is \( -7(1) + 9(7) = -7 + 63 = 56 \). Therefore, \( (AB)C = \begin{bmatrix} 26 & 90 \ 13 & 56 \end{bmatrix} \).

Step 4: Multiply B and C

To find \( BC \), multiply matrix \( B \) by matrix \( C \). The element in the first row and first column is \( -2(2) + 3(3) = -4 + 9 = 5 \), and for the first row and second column, it's \( -2(1) + 3(7) = -2 + 21 = 19 \). For the second row and first column, it's \( -1(2) + 2(3) = -2 + 6 = 4 \), and for the second row and second column, it's \( -1(1) + 2(7) = -1 + 14 = 13 \). Thus, \( BC = \begin{bmatrix} 5 & 19 \ 4 & 13 \end{bmatrix} \).

Step 5: Multiply A with (BC)

Now, multiply \( A \) with \( (BC) \). The first element is \( 2(5) + 4(4) = 10 + 16 = 26 \), the second element is \( 2(19) + 4(13) = 38 + 52 = 90 \). The third element is \( 5(5) + (-3)(4) = 25 - 12 = 13 \), and the fourth element is \( 5(19) + (-3)(13) = 95 - 39 = 56 \). Therefore, \( A(BC) = \begin{bmatrix} 26 & 90 \ 13 & 56 \end{bmatrix} \).

Step 6: Compare (AB)C and A(BC)

Upon comparison, \( (AB)C = \begin{bmatrix} 26 & 90 \ 13 & 56 \end{bmatrix} \) and \( A(BC) = \begin{bmatrix} 26 & 90 \ 13 & 56 \end{bmatrix} \) are equal, confirming that \( (AB)C = A(BC) \).

Key Concepts

Associative Property of Matrices

When we talk about the associative property in the context of matrices, we're referring to a property that allows us to group operations in different ways without changing the overall result. This property is quite powerful in linear algebra, as it facilitates complex matrix calculations by letting us rearrange matrices to suit the problem we're solving. In mathematical terms, the associative property is expressed as \((AB)C = A(BC)\), where \(A\), \(B\), and \(C\) are matrices.

Notice how the order of matrix multiplication remains crucial here, as matrices are not generally commutative. This means that \(AB eq BA\) in most cases. However, the associative property gives us flexibility in choosing which matrices to multiply first, thus simplifying calculations and reducing computation errors.

In the exercise, applying the associative property confirms that no matter which multiplication we perform first, the result remains the same. This can be quite handy in computational tasks or when optimizing algorithms.

Matrix Operations

Matrix operations are fundamental tasks in linear algebra, used to manipulate and solve sets of equations or model complex systems. The primary operations include addition, subtraction, and most importantly, multiplication.

• Matrix addition and subtraction: These operations are performed element-wise and are only possible between matrices of the same size.
• Matrix multiplication: Unlike addition, matrix multiplication involves a more complex process. We multiply rows by columns and sum the results to obtain the elements of the new matrix.

Matrix multiplication is particularly important because it's not only essential for transforming vectors and solving linear systems but also because it's a foundation of more complex operations in linear transformations and systems of equations.

In the provided example, we multiplied given matrices as per the rules of matrix multiplication: row elements from the first matrix are paired and multiplied with corresponding column elements from the second matrix, summing the results to form each element of the product matrix. This method was crucial to demonstrating the associative property effectively.

Matrix Algebra

Matrix algebra is a branch of mathematics that focuses on the study of matrices and their role in transforming and solving algebraic equations and linear systems. It provides a framework for dealing with vectors, linear transformations, and data structures efficiently.

Within matrix algebra, you apply various operations and properties such as the associative property, invertibility, transposition, and determinants. Understanding these principles not only aids in solving practical problems but also in modeling physical, economic, and computational scenarios.

• Matrix algebra extends basic algebra to higher dimensions, where each item in the system is manipulated as a linear combination of others.
• It forms the basis of linear equations representation and is instrumental in fields such as computer graphics, machine learning, and econometrics.

In our exercise, the use of matrix algebra allowed us to represent and solve the equation \((AB)C = A(BC)\) systematically. By running through these algebraic steps, we can confidently validate mathematical models and systems, an invaluable skill in both academic and professional scenarios.