Problem 103 - Precalculus

Question

$$\text { For } A=\left[\begin{array}{ll}1 & 1 \\\1 & 1\end{array}\right] \text { find } A, A^{2}, A^{3}, \ldots . . \text { What is } A^{n} ?$$

Step-by-Step Solution

Answer

\(A^n = \begin{bmatrix} 2^{(n-1)} & 2^{(n-1)} \\ 2^{(n-1)} & 2^{(n-1)} \end{bmatrix}\).

Step 1: Understanding Matrix Multiplication

To find the powers of matrix \(A\), we need to multiply the matrix by itself. Matrix multiplication involves taking the dot product of rows and columns. Remember, each element in the resulting matrix after multiplication is the sum of products of corresponding elements from rows and columns between two matrices.

Step 2: Compute \(A^2\)

Given \( A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix} \), we compute \(A^2 = A \cdot A\). Calculate each element: - First row, first column: \(1 \times 1 + 1 \times 1 = 2\);- First row, second column: \(1 \times 1 + 1 \times 1 = 2\);- Second row, first column: \(1 \times 1 + 1 \times 1 = 2\);- Second row, second column: \(1 \times 1 + 1 \times 1 = 2\).The resulting matrix is \(A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\).

Step 3: Compute \(A^3\)

Now compute \(A^3 = A^2 \cdot A\). Since \(A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\), calculate elements:- (1,1): \(2 \times 1 + 2 \times 1 = 4\);- (1,2): \(2 \times 1 + 2 \times 1 = 4\);- (2,1): \(2 \times 1 + 2 \times 1 = 4\);- (2,2): \(2 \times 1 + 2 \times 1 = 4\).So, \(A^3 = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}\).

Step 4: Identify the Pattern

Notice that the entries of the matrix double with each power. Observing the pattern from \(A^1\) to \(A^3\), each entry is equal to \(2^{(n-1)}\). Thus, for \(A^n\), the matrix will be \[A^n = \begin{bmatrix} 2^{(n-1)} & 2^{(n-1)} \ 2^{(n-1)} & 2^{(n-1)} \end{bmatrix}\].

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra. It allows us to combine matrices, resulting in another matrix that expresses complex relationships between the data sets represented by each factor. In order to multiply two matrices, follow these steps:

• Compatibility: Ensure the matrices are conformable. The number of columns in the first matrix must match the number of rows in the second matrix.
• Dot Product: For each element in the result matrix, calculate the dot product of the corresponding row from the first matrix and the column from the second matrix. The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
• Resultant Element: Place the resultant value from the dot product into the appropriate position in the new matrix corresponding to the row and column used in the calculation.

In our exercise, to find the power of matrix \(A\), we multiply \(A\) with itself several times. For example, in calculating \(A^2\), we multiply \(A\) by \(A\), computing the dot product to find each element of the resultant matrix.

Matrix Patterns

When exploring powers of matrices, recognizing patterns in the resultant matrices can greatly simplify complex computation tasks. In the exercise provided, observe how the matrix evolves as we calculate higher powers:

• Start with matrix \(A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\).


• Compute \(A^2 = \begin{bmatrix} 2 & 2 \ 2 & 2 \end{bmatrix}\), and see that each element doubles.


• Similarly, compute \(A^3 = \begin{bmatrix} 4 & 4 \ 4 & 4 \end{bmatrix}\), where once again each element doubles compared to \(A^2\).

This indicates a pattern where each element in the matrix after multiplication is \(2^{n-1}\), where \(n\) is the power of the matrix. By identifying such patterns, we can efficiently predict matrix entries for any given power without repeating exhaustive calculations.

Dot Product

The dot product is a fundamental component of matrix multiplication, which involves multiplying corresponding elements and summing up the results. It is crucial for understanding how matrix elements interact as they undergo multiplication. To perform a dot product:

• Pair Elements: Take the elements of one row from the first matrix and the corresponding elements from a column in the second matrix.
• Multiply and Sum: Multiply each pair together and then add these products to obtain a single scalar value.

This process is repeated for every row and column pairing in the matrices being multiplied. In our exercise, for matrix \(A = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\), each dot product calculation like \(1 \times 1 + 1 \times 1 = 2\) helps derive the entries of matrices \(A^2\) and \(A^3\). The repetition of this for each matrix operation means recognizing the pattern in these scalars becomes a powerful tool for efficiently handling complex calculations.