Problem 33 - 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. $$A^{3}$$

Step-by-Step Solution

Answer

\(A^3 = \begin{bmatrix} 8 & -325 \\ 0 & 343 \end{bmatrix}\)

Step 1: Understand the Operation

The problem asks us to find \(A^3\), which means multiplying matrix \(A\) by itself three times: \(A \times A \times A\). Matrix \(A\) is a 2x2 matrix.

Step 2: Multiply Matrix A by Itself Once

First, we calculate \(A^2\) by multiplying matrix \(A\) by itself. Matrix \(A\) is \(\begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\). Use the matrix multiplication rule: \( (AB)_{ij} = \sum_{k} A_{ik}B_{kj} \).\[A^2 = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 2 \times 2 + (-5) \times 0 & 2 \times (-5) + (-5) \times 7 \ 0 \times 2 + 7 \times 0 & 0 \times (-5) + 7 \times 7 \end{bmatrix} = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix}\]

Step 3: Multiply A^2 by A to get A^3

Now, multiply \(A^2\) with \(A\) to get \(A^3\).\[A^3 = \begin{bmatrix} 4 & -45 \ 0 & 49 \end{bmatrix} \times \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix} = \begin{bmatrix} 4 \times 2 + (-45) \times 0 & 4 \times (-5) + (-45) \times 7 \ 0 \times 2 + 49 \times 0 & 0 \times (-5) + 49 \times 7 \end{bmatrix} = \begin{bmatrix} 8 & -325 \ 0 & 343 \end{bmatrix}\]

Step 4: Confirm Calculation

Verify that all operations were performed correctly: double check each of the multiplication and addition operations. Ensure the final matrix \(A^3\) is accurate.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves two matrices, where the elements are combined using a specific rule. The process requires taking the dot product of rows from the first matrix, often called matrix "A", with columns from the second matrix, known as matrix "B".

• Dot Product: This operation sums the products of corresponding elements of the row and column.
• Element Indexing: In a resulting matrix "C", the element at position \(C_{ij}\) is calculated as \(C_{ij} = \sum_{k} A_{ik} B_{kj}\).

Each element in the resulting matrix is a sum of products, making the placement of numbers crucial for accurate computation. Accuracy is achieved by carefully aligning the indices and performing each multiplication and addition sequentially. Proper mastery of matrix multiplication is vital as it serves as the basis for more complex operations in linear algebra, such as matrix exponentiation.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices, having two rows and two columns. Each element in this matrix can be denoted as part of a four-element structure:

• a, b - represents the first row.
• c, d - represents the second row.

In general terms, a 2x2 matrix can be written as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]These matrices are important in solving numerous problems due to their simplicity and ease of calculation. For example, the identity matrix and rotational matrices often appear in 2x2 forms.
When a 2x2 matrix is raised to a power, as seen in determining \(A^3\), you repeatedly multiply the matrix by itself using matrix multiplication rules. This maintains the 2x2 structure while potentially altering the numeric values.

Algebraic Operations

Algebraic operations on matrices encompass a variety of techniques including addition, subtraction, multiplication, and more. In the context of matrices, these operations follow specific rules that ensure consistency and correctness.

• Addition/Subtraction: This involves element-wise operations between two matrices of the same dimensions. If the dimensions do not match, these operations cannot be performed.
• Scalar Multiplication: Each element of the matrix is multiplied by the scalar, which is a simple number.

In the case of matrix exponentiation, as requested by the exercise, multiplication is the key operation. To compute something like \(A^3\), you perform successive matrix multiplications, ensuring to follow row-column rules meticulously. This rigorous procedure enhances understanding of not just the operations, but also the properties and behavior of matrix expressions.