Problem 56 - College Algebra

Question

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. $$ \mathrm{B}=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right] $$ $$ B^{3} $$

Step-by-Step Solution

Answer

\( B^3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \).

Step 1: Understanding Matrix Multiplication

To find the cube of the matrix \( B \), we must calculate \( B \times B \times B \). Begin by understanding that matrix multiplication involves the dot product of rows and columns.

Step 2: Calculate \( B^2 \)

First, find \( B^2 = B \times B \). Multiply the first matrix by the second by moving row by column as follows:Calculate the element in the first row, first column: \((1 \times 1 + 0 \times 0 + 0 \times 0) = 1\)Calculate the element in the first row, second column: \((1 \times 0 + 0 \times 0 + 0 \times 1) = 0\)Calculate the element in the first row, third column: \((1 \times 0 + 0 \times 1 + 0 \times 0) = 0\)Calculate the element in the second row, first column: \((0 \times 1 + 0 \times 0 + 1 \times 0) = 0\)Calculate the element in the second row, second column: \((0 \times 0 + 0 \times 0 + 1 \times 1) = 1\)Calculate the element in the second row, third column: \((0 \times 0 + 0 \times 1 + 1 \times 0) = 0\)Calculate the element in the third row, first column: \((0 \times 1 + 1 \times 0 + 0 \times 0) = 0\)Calculate the element in the third row, second column: \((0 \times 0 + 1 \times 0 + 0 \times 1) = 0\)Calculate the element in the third row, third column: \((0 \times 0 + 1 \times 1 + 0 \times 0) = 1\)Thus, \( B^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \).

Step 3: Calculate \( B^3 \)

Now, use \( B^2 \) to find \( B^3 = B^2 \times B \), noting that \( B^2 \) is the identity matrix:Calculate the elements:- First row, first column: \((1 \times 1 + 0 \times 0 + 0 \times 0) = 1\)- First row, second column: \((1 \times 0 + 0 \times 0 + 0 \times 1) = 0\)- First row, third column: \((1 \times 0 + 0 \times 1 + 0 \times 0) = 0\)- Second row, first column: \((0 \times 1 + 1 \times 0 + 0 \times 0) = 0\)- Second row, second column: \((0 \times 0 + 1 \times 0 + 0 \times 1) = 0\)- Second row, third column: \((0 \times 0 + 1 \times 1 + 0 \times 0) = 1\)- Third row, first column: \((0 \times 1 + 0 \times 0 + 1 \times 0) = 0\)- Third row, second column: \((0 \times 0 + 0 \times 0 + 1 \times 1) = 1\)- Third row, third column: \((0 \times 0 + 0 \times 1 + 1 \times 0) = 0\)This results in \( B^3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \).

Step 4: Conclusion: Matrix Operation Result

Thus, the operation yields the matrix \( B^3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \). This calculation shows that repeating the permutations applied by \( B \) twice more results in the original permutation being preserved.

Key Concepts

Matrix Powers

The concept of matrix powers is all about raising a matrix to a certain exponent, much like you would with numbers. When given a matrix, let's call it \( B \), it's about understanding how to multiply it by itself a specific number of times. For instance, \( B^3 \) means multiplying the matrix \( B \) by itself three times: \( B \times B \times B \).

In order to calculate matrix powers, we rely on matrix multiplication rules, which involve multiplying the rows of the first matrix by the columns of the second. This approach can provide useful results, such as reducing to an identity matrix or transforming into a new matrix with distinct properties.

Matrix powers appear often in various applications such as computer graphics, probability, and even quantum mechanics, where transformations need to be repeated several times. Understanding this concept helps in simplifying problems where repeated operations are involved.

Identity Matrix

The identity matrix is a fundamental part of linear algebra, behaving much like the number 1 in ordinary arithmetic. It’s a special square matrix, usually denoted as \( I \), which has ones on the diagonal and zeroes elsewhere. For a 3x3 matrix, the identity matrix looks like this:

• \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \)

When a matrix is multiplied by the identity matrix, it remains unchanged. This means \( I \times B = B \) or \( B \times I = B \). In the exercise provided, when calculating \( B^2 \), we effectively reach the identity matrix. This tells us that multiplying \( B \) once more by \( B^2 \) yields \( B \) itself, because \( B^2 \) equals the identity matrix.

In many ways, the identity matrix is crucial because it acts as a mirror reflecting elements of the original matrix back upon multiplication. It's widely used when dealing with inversions and transformations to maintain the structure of matrices.

Matrix Operations

Matrix operations are the various computations you can perform with matrices, including addition, subtraction, and, as seen in the exercise, multiplication.
Matrix multiplication is a slightly more intricate operation compared to scalar multiplication or addition because it involves calculating the dot product of rows and columns from two matrices paired together.

Here's a quick look at some key properties:

• Associative: \( (A \times B) \times C = A \times (B \times C) \)
• Distributive: \( A \times (B + C) = A \times B + A \times C \)
• Not Commutative: \( A \times B eq B \times A \), in general.

Understanding these operations allows us to work through complex transformations and solve multi-step problems effectively, such as raising a matrix to a power or applying series of transformations. It's all about combining matrices or modifying them to get from one point to another, calculating intermediate steps, like finding \( B^2 \) before \( B^3 \), or preserving qualities such as a permutation when required.