Problem 55 - College Algebra

Question

For the following exercises, use the matrix below to perform the indicated operation on the given matrix. $$ \mathrm{B}=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right] $$ $$ B^{2} $$

Step-by-Step Solution

Answer

The square of matrix B, \( B^2 \), is the identity matrix.

Step 1: Understand the Operation

The exercise asks to compute the square of matrix \( B \), which implies we need to multiply matrix \( B \) by itself.

Step 2: Set up the Multiplication

Write out the matrix multiplication of \( B \) with \( B \):\[B = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}\]\[B \times B = \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{bmatrix}\]

Step 3: Multiply the Matrices

Perform matrix multiplication, computing each element of the resulting matrix.- First row: - \( (1*1 + 0*0 + 0*0) = 1 \) - \( (1*0 + 0*0 + 0*1) = 0 \) - \( (1*0 + 0*1 + 0*0) = 0 \)- Second row: - \( (0*1 + 0*0 + 1*0) = 0 \) - \( (0*0 + 0*0 + 1*1) = 1 \) - \( (0*0 + 0*1 + 1*0) = 0 \)- Third row: - \( (0*1 + 1*0 + 0*0) = 0 \) - \( (0*0 + 1*0 + 0*1) = 0 \) - \( (0*0 + 1*1 + 0*0) = 1 \) Thus, the resulting matrix is: \[B^2 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]

Step 4: Verify the Result

Check that each element of the resulting matrix \( B^2 \) is correctly derived from the multiplication. The resulting matrix is the identity matrix, which confirms our calculations are correct.

Key Concepts

Matrix Squaring

Matrix squaring refers to the process of multiplying a matrix by itself. In this operation, we take a square matrix like matrix \( B \) from the exercise and perform matrix multiplication where \( B \) is multiplied by \( B \) again. This results in a new matrix referred to as \( B^2 \). Matrix squaring is an essential concept in linear algebra and has practical applications in engineering and computer science.
If you're new to matrices, remember that this isn't just multiplying numbers directly. Instead, each element in the resulting matrix \( B^2 \) is obtained by taking the dot product of corresponding rows and columns from the matrices being multiplied. For the given matrix

• Calculate each element of the square by focusing on each row of the first matrix and each column of the second matrix.
• Sum the products of corresponding elements.

In this exercise, the matrix \( B^2 \) resulted in the identity matrix, which we will explore next.

Identity Matrix

An identity matrix is a special type of square matrix that plays an important role in matrix operations. It is characterized by having all diagonal elements equal to 1 and all off-diagonal elements equal to 0. In mathematical terms, an identity matrix of size \( n \times n \) is expressed as \( I_n \). The property of the identity matrix is analogous to multiplying numbers by 1. For any matrix \( A \), multiplying by an identity matrix leaves \( A \) unchanged: \( A \times I_n = A \).
In the exercise above, squaring the matrix \( B \) gives us an identity matrix:

• First row only affects the first column, resulting in the first diagonal element as 1.
• The second row, affecting the second column, results in the second diagonal element as 1.
• The third row, similarly affects the third column, keeping the third diagonal element as 1.

This showcases the importance of understanding the identity matrix and how matrix operations can yield such significant outcomes.

Matrix Operations

Matrix operations encompass a range of procedures including addition, subtraction, and multiplication of matrices. These operations form the backbone of many mathematical procedures and applications, particularly in fields like physics, computer graphics, and economics.
Consider the matrix multiplication that took place in the solution. When multiplying matrices, ensure:

• The number of columns in the first matrix equals the number of rows in the second matrix for the multiplication to be defined.
• The resulting matrix has the same number of rows as the first matrix, and the same number of columns as the second matrix.

Understanding matrix operations requires practice to get the hang of concepts like dot products and row-and-column correspondences. In this exercise, matrix \( B \) was organized in such a way that its square resulted in the identity matrix. By practicing matrix operations, you can unravel more complex problems and appreciate how matrices are used to model real-world systems.