Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. \(B=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]\) $$ B^{4} $$

Step-by-Step Solution

Answer

The matrix \(B^4\) is the identity matrix \(I\).

Step 1: Understand Operation Type

We are required to calculate the fourth power of the given matrix, which involves matrix multiplication. This means multiplying the matrix by itself multiple times.

Step 2: Matrix Multiplication Review

Matrix multiplication is associative. To compute \(B^4\), we will multiply the matrix \(B\) by itself, starting from \(B^2 = B \times B\), then calculate \(B^3 = B^2 \times B\), and finally, \(B^4 = B^3 \times B\).

Step 3: Calculate \(B^2\)

First, compute the product \(B \times B\):\[B = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}\]\[B \times B = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]Notice that \(B^2\) is the identity matrix \(I\).

Step 4: Calculate \(B^3\)

Next, multiply \(B^2\) by \(B\):Since \(B^2\) is the identity matrix, \(B^3 = I \times B = B\), so it remains:\[B^3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}\]

Step 5: Calculate \(B^4\)

Finally, multiply \(B^3\) by \(B\):\[B^4 = B^3 \times B = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]Thus, \(B^4\) is again the identity matrix.

Key Concepts

Matrix Powers

Matrix powers refer to the process of multiplying a matrix by itself multiple times. When a matrix is raised to a power, it indicates a repeated multiplication of the matrix by itself. For example, when computing the fourth power of a matrix, denoted as \(B^4\), we multiply the matrix \(B\) by itself three additional times after the initial matrix is considered. Understanding how to compute a matrix power involves iteratively applying matrix multiplication.

• Start by calculating \(B^2\), meaning \(B \times B\).
• Next, find \(B^3\) by multiplying \(B^2\) by \(B\) again.
• Finally, compute \(B^4\) by performing \(B^3 \times B\).

Matrix powers are essential in many computational applications, such as solving systems of equations and transformations in physics and computer graphics.

Identity Matrix

The identity matrix plays a pivotal role in matrix operations, often acting as the multiplicative identity. For a square matrix of size \(n \times n\), the identity matrix \(I_n\) has ones along its diagonal and zeros elsewhere. When any matrix \(A\) is multiplied by the identity matrix \(I\), the result is the matrix \(A\) itself. This property is similar to multiplying a number by one.

• It's crucial in defining matrix inverse operations.
• Serves as the "starting point" in matrix multiplication series.
• Any matrix raised to the power zero equals the identity matrix.

In the example given, we observed that after calculating \(B^2\), the result was the identity matrix. This outcome simplifies successive calculations because multiplying any matrix by the identity matrix results in the original matrix itself.

Associativity in Matrix Operations

Associativity is a fundamental property of matrix multiplication, indicating that the grouping of matrices does not affect the overall product. In mathematical terms, for any matrices \(A\), \(B\), and \(C\) for which the products are defined, \((A \times B) \times C = A \times (B \times C)\).

• Associativity allows you to multiply matrices in any order within grouped products.
• It's crucial when computing higher powers of matrices, enabling a step-by-step approach.
• Helps ensure accuracy and consistency in matrix calculations.

This property was demonstrated in our matrix power calculation, as we grouped operations to compute \(B^2\), \(B^3\), and ultimately \(B^4\). Each time, the grouping of operations gave the same result, emphasizing the reliability of associativity in reducing computational complexity.