Problem 51 - Algebra and Trigonometry

Question

Powers of a Matrix Let $$ A=\left[\begin{array}{ll}{1} & {1} \\ {0} & {1}\end{array}\right] $$ Calculate \(A^{2}, A^{3}, A^{4}, \ldots\) until you detect a pattern. Write a general formula for \(A^{n} .\)

Step-by-Step Solution

Answer

The general formula for \(A^n\) is \(A^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}\).

Step 1: Calculate A^2

First, we need to multiply matrix A by itself to find \(A^2\). Given \(A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}\), calculate: \[A^2 = A \times A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} (1 \cdot 1 + 1 \cdot 0) & (1 \cdot 1 + 1 \cdot 1) \ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \]

Step 2: Calculate A^3

Multiply \(A^2\) with \(A\) to get \(A^3\): \[A^3 = A^2 \times A = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} (1 \cdot 1 + 2 \cdot 0) & (1 \cdot 1 + 2 \cdot 1) \ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 & 3 \ 0 & 1 \end{bmatrix} \]

Step 3: Calculate A^4

Similarly, multiply \(A^3\) with \(A\) to get \(A^4\): \[A^4 = A^3 \times A = \begin{bmatrix} 1 & 3 \ 0 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} (1 \cdot 1 + 3 \cdot 0) & (1 \cdot 1 + 3 \cdot 1) \ (0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 1 & 4 \ 0 & 1 \end{bmatrix} \]

Step 4: Identify the Pattern

Observe the results for \(A^2, A^3, A^4\):\(A^2 = \begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}\), \(A^3 = \begin{bmatrix} 1 & 3 \ 0 & 1 \end{bmatrix}\), \(A^4 = \begin{bmatrix} 1 & 4 \ 0 & 1 \end{bmatrix}\). The upper right entry increases by 1 each time, equal to the exponent n.Therefore, \(A^n = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}\).

Step 5: General Formula

Using the pattern identified, the general formula for \(A^n\) is: \[A^n = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra. Understanding it is essential to any work with matrices, including calculating powers of a matrix. When multiplying two matrices, it’s important to note that matrix multiplication is not the same as multiplying individual numbers. Instead, it involves the dot product of rows from the first matrix and columns from the second.

• To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second.
• The result is a new matrix where each entry is derived from multiplying corresponding rows and columns.

Take for example our matrix \(A = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix}\). To calculate \(A^2\), you multiply \(A\) to itself. The resulting matrix is derived from:

• Top-left: \((1\cdot1 + 1\cdot0) = 1\)
• Top-right: \((1\cdot1 + 1\cdot1) = 2\)
• Bottom-left: \((0\cdot1 + 1\cdot0) = 0\)
• Bottom-right: \((0\cdot1 + 1\cdot1) = 1\)

The resulting matrix is \(\begin{bmatrix} 1 & 2 \ 0 & 1 \end{bmatrix}\). This process repeats for higher powers.

General Formula for Matrices

Finding a pattern or formula for a matrix raised to a power can make calculations more efficient. When we observe how matrices multiply, patterns often appear. These patterns are utilized to formulate general expressions for any power.
In the exercise, you continued multiplying matrix \(A\), noting the results. Each multiplication revealed an emerging pattern, specifically in the top-right element of these products. Structure this pattern into a general formula.

• Every multiplication adds 1 to the top-right element.
• The top-left and bottom-right remain constant as 1.
• The bottom-left stays 0 throughout.

Thus, for matrix \(A\) raised to the nth power, you found the result could be anticipated: \(A^n = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix}\). Recognizing such structures simplifies calculations drastically, as instead of calculating each multiplication, the pattern gives immediate answers.

Linear Algebra Concepts

Linear algebra extends beyond mere number calculations, using matrices and vectors to represent and solve systems of equations. Central to this are concepts like matrix operations, such as addition, scalar multiplication, and matrix multiplication. Another vital idea is the manipulation of matrices to achieve specific forms or solutions.

• Understanding matrix operations enables solving complex problems in physics and computer science.
• Powers of matrices can model repetitive processes or transformations in systems.
• Patterns within matrix operations can often simplify complex calculations, making them manageable.

In the exercise where matrix \(A\) is raised to successive powers, the focus is on recognizing matrix behavior under repeated operations. Realizing how certain entries change and others remain constant provides deep insights into the matrix's structure and potential applications. Such insights are essential for advancing in fields that rely on linear algebra, like engineering or data science.