Problem 57 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

You are asked to find the fourth power of matrix \( \mathrm{B} \). This means we need to multiply the matrix \( \mathrm{B} \) by itself four times: \( \mathrm{B} \times \mathrm{B} \times \mathrm{B} \times \mathrm{B} \).

Step 2: Multiply B by Itself Once (Find B^2)

First, calculate \( B^2 \) by multiplying \( \mathrm{B} \) by itself: \[B \times B = \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right] \times \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\]This result is the identity matrix.

Step 3: Multiply the Result by B Once More (Find B^3)

Next, we find \( B^3 \) by multiplying \( B^2 \) (which is the identity matrix) by \( \mathrm{B} \):\[B^2 \times B = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right] \times \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right] = \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right]\]The result is \( \mathrm{B} \) again.

Step 4: Multiply the Result by B Once More (Find B^4)

Finally, calculate \( B^4 \) by multiplying \( B^3 \) (which is \( \mathrm{B} \)) by \( \mathrm{B} \):\[B^3 \times B = \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right] \times \left[\begin{array}{lll}{1} & {0} & {0} \ {0} & {0} & {1} \ {0} & {1} & {0}\end{array}\right] = \left[\begin{array}{ccc}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right]\]Again, the result is the identity matrix.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves taking two matrices and producing a third matrix. However, it's not as straightforward as multiplying numbers. The process follows specific rules:

• The number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be possible.
• The resulting matrix has dimensions that are determined by the rows of the first matrix and the columns of the second matrix.
• For each element in the resulting matrix, corresponding elements are taken from the row of the first matrix and the column of the second matrix, multiplied, and summed.

In the context of the exercise, multiplying matrix \( B \) by itself is an example of repeated matrix multiplication, which we pursue to determine higher powers of \( B \). Each multiplication follows these rules: for example, calculating \( B^2 \) involves explicitly multiplying \( B \) with itself once, adhering to the alignment and summation of elements as described.

Identity Matrix

The identity matrix is a special type of matrix with ones on the diagonal and zeros elsewhere. In matrix algebra, it's the equivalent of the number 1 in ordinary multiplication. When you multiply any matrix by an identity matrix (of compatible dimensions), the original matrix remains unchanged.

• The identity matrix is denoted as \( I \).
• In terms of dimensions, an \( n \times n \) identity matrix is noted as \( I_n \), where all diagonal elements are 1 and all other elements are 0.
• For a matrix \( A \), multiplying it with the identity matrix produces the result: \( A \times I_n = A \).

During the original exercise, upon computing \( B^2 \), the resulting product is an identity matrix. This property plays a crucial role, especially in matrix exponentiation, where powers of matrices are taken.

Powers of Matrices

Powers of matrices involve multiplying a matrix by itself repeatedly. The notion is similar to raising numbers to a power in regular arithmetic. If you have a matrix \( B \), then its powers are defined as:

• \( B^1 = B \),
• \( B^2 = B \times B \),
• \( B^3 = B^2 \times B \), and so on.

One important rule is that if a matrix becomes an identity matrix during these multiplications, future multiplications will also result in the cycle repeating back to the original matrix or another predictable pattern.

In this exercise, after finding \( B^2 \) as the identity matrix, every subsequent multiplication simply reproduces the process. Consequently, \( B^4 \), as calculated, turns out to be the identity matrix again. Hence, understanding powers of matrices can simplify calculations, reducing the need for repetitive multiplication if certain patterns, like returning to the identity matrix, emerge.