Problem 33 - Algebra and Trigonometry

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows. $$A=\left[\begin{array}{rr}{2} & {-5} \\ {0} & {7}\end{array}\right] \quad B=\left[\begin{array}{rrr}{3} & {\frac{1}{2}} & {5} \\ {1} & {-1} & {3}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-\frac{5}{2}} & {0} \\ {0} & {2} & {-3}\end{array}\right]$$ $$\begin{array}{l}{D=\left[\begin{array}{lll}{7} & {3}\end{array}\right]} & {E=\left[\begin{array}{lll}{1} \\ {1} \\ {2} \\ {0}\end{array}\right]} \\\ {F=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] \quad G=\left[\begin{array}{rrr}{5} & {-3} & {10} \\\ {6} & {1} & {0} \\ {-5} & {2} & {2}\end{array}\right]}\end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$ A^{3} $$

Step-by-Step Solution

Answer

\( A^3 = \begin{bmatrix} 8 & -335 \\ 0 & 343 \end{bmatrix} \)

Step 1: Understand the operation

The operation requires us to find the cube of matrix \( A \), represented as \( A^3 \). This entails multiplying matrix \( A \) by itself twice: \( A \times A \times A \).

Step 2: Multiply Matrix A by Itself

Matrix \( A \) is a \( 2 \times 2 \) matrix:\[ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]First, calculate \( A^2 = A \times A \):\[ A \times A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]The result is:1. First row, first column: \( (2 \times 2) + (-5 \times 0) = 4 \)2. First row, second column: \( (2 \times -5) + (-5 \times 7) = -10 - 35 = -45 \)3. Second row, first column: \( (0 \times 2) + (7 \times 0) = 0 \)4. Second row, second column: \( (0 \times -5) + (7 \times 7) = 49 \)Thus, \( A^2 = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \).

Step 3: Multiply A^2 by A

Next, calculate \( A^3 = A^2 \times A \):\[ \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \]The result is:1. First row, first column: \( (4 \times 2) + (-45 \times 0) = 8 \)2. First row, second column: \( (4 \times -5) + (-45 \times 7) = -20 - 315 = -335 \)3. Second row, first column: \( (0 \times 2) + (49 \times 0) = 0 \)4. Second row, second column: \( (0 \times -5) + (49 \times 7) = 343 \)Thus, \( A^3 = \begin{bmatrix} 8 & -335 \ 0 & 343 \end{bmatrix} \).

Key Concepts

Matrix Operations

Matrix operations encompass a variety of mathematical procedures that can be performed on matrices, such as addition, subtraction, and multiplication. In this exercise, the primary focus is on matrix multiplication.
Matrix multiplication involves the combination of two matrices to produce a new matrix through specific arithmetic rules. It is crucial to note that matrix multiplication is not commutative—meaning that the order in which matrices are multiplied matters. For example, multiplying matrix A by matrix B, denoted as \( A \cdot B \), does not typically yield the same result as multiplying matrix B by matrix A, \( B \cdot A \).
When multiplying matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Each element in the resulting matrix is computed as the sum of the products of the corresponding entries from the rows of the first matrix and the columns of the second matrix. Be mindful of these rules to accurately perform matrix multiplication.

2x2 Matrices

A 2x2 matrix is a specific type of matrix that has two rows and two columns. They are simple yet powerful in showcasing basic matrix operations due to their manageable size. In this exercise, the matrix we work with is \( A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \).
Working with a 2x2 matrix typically involves using straightforward algebraic operations, such as addition, subtraction, and multiplication, which follow particular rules.

• Matrix Addition and Subtraction: Only matrices of the same dimension can be added or subtracted. Add or subtract the corresponding entries from each matrix.
• Matrix Multiplication: Multiply the elements of the rows of the first matrix by the elements of the columns of the second matrix and sum them.

Due to their structure, performing operations on 2x2 matrices serves as a great way to build foundational knowledge of matrix algebra. They also appear frequently in physics, engineering, and computer science, making them a practical learning tool.

Exponentiation of Matrices

Exponentiation of matrices involves multiplying a matrix by itself a given number of times. In this exercise, we calculated the cube of matrix A, denoted as \( A^3 \). This involves two steps: first, calculate \( A^2 \), then multiply \( A^2 \) by A.
This process is vital in various applications such as solving linear differential equations and in systems theory. Calculating higher powers of matrices allows us to predict the behavior of a system over time.

• Step 1: Find \( A^2 \) by multiplying matrix \( A \) by itself.
• Step 2: Compute \( A^3 \) by multiplying \( A^2 \) with \( A \) again.

Remember that matrix exponentiation is only straightforward for square matrices (such as 2x2 matrices) and entails basic matrix multiplication principles. The result provides insightful data about the power of repeated applications of transformations described by a matrix.