Problem 56 - Algebra and Trigonometry

Question

Powers of a Matrix \(\quad\) 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: Compute \(A^2\)

We begin by multiplying matrix \(A\) by itself. Given \(A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\), compute \(A^2 = A \times A\):\[A^2 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 \times 1 + 1 \times 1 & 1 \times 1 + 1 \times 1 \ 1 \times 1 + 1 \times 1 & 1 \times 1 + 1 \times 1 \end{bmatrix} = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\].

Step 2: Compute \(A^3\)

Now, multiply the computed \(A^2\) by \(A\) to find \(A^3\):\[A^3 = A^2 \times A = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times 1 + 2 \times 1 & 2 \times 1 + 2 \times 1 \ 2 \times 1 + 2 \times 1 & 2 \times 1 + 2 \times 1 \end{bmatrix} = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}\].

Step 3: Compute \(A^4\)

Multiply \(A^3\) by \(A\) to find \(A^4\):\[A^4 = A^3 \times A = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} = \begin{bmatrix} 4 \times 1 + 4 \times 1 & 4 \times 1 + 4 \times 1 \ 4 \times 1 + 4 \times 1 & 4 \times 1 + 4 \times 1 \end{bmatrix} = \begin{bmatrix} 8 & 8 \ 8 & 8 \end{bmatrix}\].

Step 4: Identify the Pattern

Observe the pattern in the results: \(A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\), \(A^3 = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}\), \(A^4 = \begin{bmatrix} 8 & 8 \ 8 & 8 \end{bmatrix}\). Notice that the elements of the matrices are powers of 2. For \(A^n\), each element is \(2^{n-1}\).

Step 5: Write the General Formula

Based on the pattern, for any positive integer \(n\), the general formula for matrix power \(A^n\) is:\[A^n = 2^{n-1} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\].

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation for matrices, involving the rows of the first matrix and the columns of the second. When multiplying two matrices, you must ensure that the number of columns in the first matrix matches the number of rows in the second. For example, consider matrices \( A \) and \( B \). The element in the first row and first column of the resulting matrix is determined by multiplying the elements of the first row of \( A \) with the elements of the first column of \( B \), and summing the products.
In our exercise, we start with matrix \( A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \). To compute \( A^2 \), we multiply \( A \times A \) as follows:

• Compute the element in the first row, first column: \( 1 \times 1 + 1 \times 1 = 2 \).
• Compute the element in the first row, second column in the same way: \( 1 \times 1 + 1 \times 1 = 2 \).
• The same calculation applies for the second row, resulting in all elements being 2.

Therefore, \( A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix} \). This step is then repeated to find \( A^3 \) and \( A^4 \), continuing the pattern through matrix multiplication.

Pattern Recognition in Matrices

Recognizing patterns in matrices is essential for finding solutions to repeated operations. By observing repetitive structures, you identify a consistent rule or formula that simplifies calculations. In our example, creating powers of matrix \( A \), we detect an emerging pattern as we increase the power:

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

Each matrix has identical elements, which are powers of 2. The pattern here indicates that as \( n \) increases, the elements become \( 2^{n-1} \). Identifying such a pattern is crucial when finding a formula for larger powers without explicit multiplication, as it demonstrates exponential growth within a specific framework.

General Formula for Matrix Powers

Developing a general formula for matrix powers requires understanding the pattern noticed in the matrix elements. For the matrix \( A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \), recognizing the pattern leads to a simplified form of any matrix power \( A^n \) without repeated computation.
The matrices produced, such as \( A^2, A^3, \) and \( A^4 \), showed that each element can be expressed as \( 2^{n-1} \). This pattern allows us to express \( A^n \) with a general formula:

• For any positive integer \( n \), \( A^n = 2^{n-1} \times \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \).

This formula is highly efficient, allowing quick determination of any power of the matrix \( A \) by understanding its exponential component. The formula reveals that no matter how large \( n \) becomes, the simplicity of the relationship helps in swiftly calculating matrix powers, eliminating the need for lengthy matrix multiplications.