Problem 32 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The resulting matrix \( A^2 \) is \( \begin{bmatrix} 4 & -45 \\ 0 & 49 \end{bmatrix} \).

Step 1: Define Matrix A

Matrix \( A \) is defined as follows: \[ A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \] It is a \( 2 \times 2 \) matrix.

Step 2: Understand the Operation

The operation \( A^2 \) implies that we need to multiply matrix \( A \) by itself.

Step 3: Perform Matrix Multiplication

To square the matrix \( A \), we calculate \( A \times A \): \[ \begin{pmatrix} 2 & -5 \ 0 & 7 \end{pmatrix} \times \begin{pmatrix} 2 & -5 \ 0 & 7 \end{pmatrix} = \begin{pmatrix} (2 \cdot 2) + (-5 \cdot 0) & (2 \cdot -5) + (-5 \cdot 7) \ (0 \cdot 2) + (7 \cdot 0) & (0 \cdot -5) + (7 \cdot 7) \end{pmatrix} \]

Step 4: Calculate Each Element

Calculate each element of the resulting matrix: - The element in the first row, first column: \( 2 \cdot 2 + (-5 \cdot 0) = 4 \)- The element in the first row, second column: \( 2 \cdot -5 + (-5 \cdot 7) = -10 - 35 = -45 \)- The element in the second row, first column: \( 0 \cdot 2 + 7 \cdot 0 = 0 \)- The element in the second row, second column: \( 0 \cdot -5 + 7 \cdot 7 = 49 \)

Step 5: Write the Resultant Matrix

The result of \( A \times A \) is: \[ A^2 = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. When we multiply two matrices, we are essentially calculating the dot products of rows and columns. Each element in the resulting matrix is calculated by taking the sum of the products of corresponding entries. Here’s how it works step-by-step:

• Consistency of Dimensions: A key requirement for matrix multiplication is that the number of columns in the first matrix matches the number of rows in the second matrix. If matrix A is of size \(m \times n\) and matrix B is \(n \times p\), the resulting matrix will be \(m \times p\).
• Dot Product: For each element in the resulting matrix, take the sum of the products of corresponding elements in the row of the first matrix and the column of the second matrix.

In the context of our exercise, we multiply matrix \(A\) by itself to get \(A^2\). Each step involves multiplying corresponding elements from the first row of \(A\) with columns in \(A\) again and summing them up. This principle ensures a systematic approach to matrix multiplication, providing correct and meaningful results consistently. It is crucial to practice these steps to build an intuitive understanding of matrix multiplication.

Matrix Squaring

Matrix squaring is a specific case of matrix multiplication where a matrix is multiplied by itself. Understanding this concept requires a solid grasp of basic matrix multiplication. Let’s break down matrix squaring step-by-step:

• Multiplication Setup: Choose a matrix, such as matrix \(A\), to square. Multiply it by itself.
• Element Calculation: Calculate each element of the resulting matrix using the dot product of the corresponding row and column. For instance, to find the element in the first row and the first column of \(A^2\), multiply each element of the first row of \(A\) by the corresponding element of the first column of \(A\) and then add them up.

Using these steps, we computed \(A^2\), which simplifies the matrix into a single product. Matrix squaring is a foundational operation in more advanced matrix functions and algorithms in mathematics and computer science.

2x2 Matrices

A 2x2 matrix is one of the simplest forms of rectangular arrays in linear algebra. Despite their simplicity, 2x2 matrices are incredibly powerful and often used in basic matrix operations. Here’s what you need to know:

• Definition: A 2x2 matrix has two rows and two columns. For example, \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\).
• Operation Simplicity: Because of their small size, operations like addition, multiplication, and finding determinants (which equals \(ad - bc\) for a 2x2 matrix) are straightforward and computationally efficient.
• Applications: 2x2 matrices can represent simple transformations in 2D space, such as rotations or scaling.

In our exercise, matrix \(A\) was a 2x2 matrix, making it an ideal candidate for operations like squaring, as they are simple yet demonstrate essential algebraic principles. These matrices are excellent for learning foundational concepts before applying them to larger matrices.