Problem 76 - A Graphical Approach to College Algebra

Question

Given \(A=\left[\begin{array}{cc}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{cc}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product if possible. $$A^{3}$$

Step-by-Step Solution

Answer

\( A^3 = \begin{bmatrix} 10 & -30 \\ 45 & -35 \end{bmatrix} \)."

Step 1: Understanding Matrix Exponentiation

Matrix exponentiation involves raising a matrix to a power, which is equivalent to multiplying the matrix by itself a certain number of times. Here, we are tasked with finding \( A^3 \), which means multiplying the matrix \( A \) by itself twice (i.e., \( A \times A \times A \)).

Step 2: Matrix Multiplication Fundamentals

To find \( A^2 = A \times A \), we must perform matrix multiplication. Matrix \( A = \begin{bmatrix} 4 & -2 \ 3 & 1 \end{bmatrix} \) is a 2x2 matrix, allowing for multiplication as the number of columns in the first matrix matches the number of rows in the second matrix. The resulting matrix will also be a 2x2 matrix.

Step 3: Compute \( A^2 \)

First, calculate \( A \times A \). Perform the multiplication - Top-left element: \((4 \cdot 4) + (-2 \cdot 3) = 16 - 6 = 10\) - Top-right element: \((4 \cdot -2) + (-2 \cdot 1) = -8 - 2 = -10\) - Bottom-left element: \((3 \cdot 4) + (1 \cdot 3) = 12 + 3 = 15\) - Bottom-right element: \((3 \cdot -2) + (1 \cdot 1) = -6 + 1 = -5\) Therefore, \( A^2 = \begin{bmatrix} 10 & -10 \ 15 & -5 \end{bmatrix} \).

Step 4: Compute \( A^3 \)

Now, multiply \( A^2 \) by \( A \) to get \( A^3 \): - Top-left element: \((10 \cdot 4) + (-10 \cdot 3) = 40 - 30 = 10\) - Top-right element: \((10 \cdot -2) + (-10 \cdot 1) = -20 - 10 = -30\) - Bottom-left element: \((15 \cdot 4) + (-5 \cdot 3) = 60 - 15 = 45\) - Bottom-right element: \((15 \cdot -2) + (-5 \cdot 1) = -30 - 5 = -35\) Hence, \( A^3 = \begin{bmatrix} 10 & -30 \ 45 & -35 \end{bmatrix} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike regular arithmetic multiplication, matrix multiplication involves multiplying rows by columns. To multiply two matrices, you calculate the dot product of the rows of the first matrix with the columns of the second matrix.

• For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.
• If matrix A is of size \(m \times n\) and matrix B is of size \(n \times p\), their product will result in a matrix of size \(m \times p\).
• The element in the resulting matrix at position \((i, j)\) is calculated by multiplying each component of the i-th row of the first matrix by the corresponding component of the j-th column of the second matrix and summing the results.

In matrix exponentiation, like calculating \(A^3\), the matrix is multiplied by itself multiple times. This involves repeated matrix multiplication. Understanding these fundamentals makes solving such problems more manageable.

2x2 Matrices

2x2 matrices are among simplest types of matrices, yet they form the building blocks for understanding more complex matrix operations. These matrices have a size of 2 rows and 2 columns. Let's break down the form:

• A 2x2 matrix is written as \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\).
• These matrices are straightforward enough to individually calculate by hand but still illustrate the typical properties and rules of matrix arithmetic.
• The determinant of a 2x2 matrix, which is a special number that can tell us many properties of the matrix, is calculated as \(ad - bc\).

In matrix exponentiation, when we have a 2x2 matrix like matrix A, we perform repeated multiplications, always ensuring our operations respect the small size and the specific calculations for each entry. This simplicity makes 2x2 matrices ideal for initial learning.

Matrix Operations

Matrix operations encompass addition, subtraction, multiplication, and more complex operations like invertibility and exponentiation. Each of these plays a critical role in various applications, from solving systems of equations to transformations in computer graphics.

• **Matrix addition and subtraction** are straightforward; it's just adding or subtracting the corresponding elements of matrices with the same dimensions.
• **Matrix multiplication** has specific requirements and procedures, as previously discussed, making it slightly more complex.
• **Matrix exponentiation** is about repeated multiplication of a matrix by itself, which can be useful in calculating powers of matrix transformations and is commonly used in algorithm optimizations like those found in computations involving Fibonacci numbers or other recursive formulas.

Understanding how these operations work individually and together allows you to see the powerful versatility matrices offer in various fields. Matrix exponentiation, specifically, challenges you to apply multiplication repeatedly, showcasing how complex computations can emerge from fundamental principles.