Problem 63 - Precalculus: Mathematics for Calculus

Question

Let $$ A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right] $$ 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 \( A^n = 2^{n-1} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \).

Step 1: Understand the Matrix and Operation

We are given the matrix \( A \) as \( \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \). The task is to calculate powers of this matrix: \( A^2, A^3, \) and so on, until we identify a pattern. This involves the multiplication of matrices.

Step 2: Calculate Square of Matrix A

To find \( A^2 \), multiply matrix \( A \) by itself. \\(A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} (1 \cdot 1 + 1 \cdot 1) & (1 \cdot 1 + 1 \cdot 1) \ (1 \cdot 1 + 1 \cdot 1) & (1 \cdot 1 + 1 \cdot 1) \end{bmatrix} = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}.\)

Step 3: Calculate Cube of Matrix A

To find \( A^3 \), use the result of \( A^2 \) and multiply by \( A \) again. \\(A^3 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} (2 \cdot 1 + 2 \cdot 1) & (2 \cdot 1 + 2 \cdot 1) \ (2 \cdot 1 + 2 \cdot 1) & (2 \cdot 1 + 2 \cdot 1) \end{bmatrix} = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}.\)

Step 4: Calculate Fourth Power of Matrix A

To find \( A^4 \), multiply the result of \( A^3 \) by \( A \) once more. \\(A^4 = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} (4 \cdot 1 + 4 \cdot 1) & (4 \cdot 1 + 4 \cdot 1) \ (4 \cdot 1 + 4 \cdot 1) & (4 \cdot 1 + 4 \cdot 1) \end{bmatrix} = \begin{bmatrix} 8 & 8 \ 8 & 8 \end{bmatrix}.\)

Step 5: Identify the Pattern

The entries of the resulting matrices double each time we multiply by \( A \). Hence, for any power \( n \), the resulting matrix \( A^n \) is \( 2^{n-1} \) times the initial matrix. Thus, \( A^n = 2^{n-1} \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike regular multiplication, matrix multiplication involves combining rows with columns. When multiplying two matrices, say matrix \( A \) with matrix \( B \), the element in the \( i^{th} \) row and \( j^{th} \) column of the result is found by taking the dot product of the \( i^{th} \) row of \( A \) and the \( j^{th} \) column of \( B \).
For example, consider the multiplication of our matrix \( A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \). To compute \( A^2 = A \times A \), we calculate:

• First row, first column: \((1\times1) + (1\times1) = 2\)
• First row, second column: \((1\times1) + (1\times1) = 2\)
• Second row, first column: \((1\times1) + (1\times1) = 2\)
• Second row, second column: \((1\times1) + (1\times1) = 2\)

Thus, \( A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\). It is crucial to ensure the dimensions of the matrices match correctly for multiplication to be possible. Both matrices \( A \) and \( B \) must have dimensions where the number of columns in \( A \) is equal to the number of rows in \( B \).

Pattern Recognition in Matrices

Recognizing patterns in matrices is a vital skill in algebra and helps predict outcomes without full calculations. Upon examining the resulting matrices \( A^2, A^3, A^4, \ldots \), we notice that the values in the matrix are consistent:

• \( A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\)
• \( A^3 = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}\)
• \( A^4 = \begin{bmatrix} 8 & 8 \ 8 & 8 \end{bmatrix}\)

We observe that each time we multiply, each element in the matrix doubles. This pattern suggests a consistent operation formula: for each additional power increase, the entries of the matrix are multiplied by 2. This pattern recognition saves us from repeating calculations and helps in quickly finding higher powers of the matrix.

General Formula for Matrix Powers

With the identified pattern, deriving a general formula for matrix powers becomes straightforward.
The initial matrix \( A \) is defined as \( \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \). From our pattern recognition, the matrix's entries double with each successive multiplication by \( A \). The formula can now be theorized as:\[A^n = 2^{n-1} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\]This formula is derived from the observation that from \( A^2 \) to \( A^n \), every power introduces an additional factor of 2. The exponent \( n-1 \) signifies this process. Hence, for any positive integer \( n \), the matrix raised to the power \( n \) results in a matrix where each term is \( 2^{n-1} \) times the corresponding term in the original matrix. This concise formula enables easy computation of higher powers of the matrix without manual multiplication each time.