Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B^{2} A^{2}\)

Step-by-Step Solution

Answer

\(B^{2} A^{2} = \begin{bmatrix} 332500 & 927500 \\ -227500 & 87500 \end{bmatrix}\). Matrix multiplication is possible.

Step 1: Calculate B squared

First, compute the square of matrix B, denoted as \(B^{2}\). Since B is a 2x2 matrix, we multiply B by itself: \[B^{2} = B \cdot B = \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix} \cdot \begin{bmatrix} 40 & 10 \ -20 & 30 \end{bmatrix}\]Calculating \(B \cdot B\): - First element: \(40 \cdot 40 + 10 \cdot (-20) = 1600 - 200 = 1400\)- Second element: \(40 \cdot 10 + 10 \cdot 30 = 400 + 300 = 700\)- Third element: \(-20 \cdot 40 + 30 \cdot (-20) = -800 - 600 = -1400\)- Fourth element: \(-20 \cdot 10 + 30 \cdot 30 = -200 + 900 = 700\)Therefore, \(B^{2} = \begin{bmatrix} 1400 & 700 \ -1400 & 700 \end{bmatrix}\).

Step 2: Calculate A squared

Now, compute the square of matrix A, denoted as \(A^{2}\). Similarly, since A is also a 2x2 matrix, we multiply A by itself: \[A^{2} = A \cdot A = \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix} \cdot \begin{bmatrix} -10 & 20 \ 5 & 25 \end{bmatrix}\]Calculating \(A \cdot A\):- First element: \((-10) \cdot (-10) + 20 \cdot 5 = 100 + 100 = 200\)- Second element: \((-10) \cdot 20 + 20 \cdot 25 = -200 + 500 = 300\)- Third element: \(5 \cdot (-10) + 25 \cdot 5 = -50 + 125 = 75\)- Fourth element: \(5 \cdot 20 + 25 \cdot 25 = 100 + 625 = 725\)Therefore, \(A^{2} = \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix}\).

Step 3: Calculate B squared by A squared

Finally, compute \(B^{2} \cdot A^{2}\). Both \(B^{2}\) and \(A^{2}\) are 2x2 matrices, so their multiplication is possible:\[B^{2} \cdot A^{2} = \begin{bmatrix} 1400 & 700 \ -1400 & 700 \end{bmatrix} \cdot \begin{bmatrix} 200 & 300 \ 75 & 725 \end{bmatrix}\]Calculating the matrix multiplication:- First element: \(1400 \cdot 200 + 700 \cdot 75 = 280000 + 52500 = 332500\)- Second element: \(1400 \cdot 300 + 700 \cdot 725 = 420000 + 507500 = 927500\)- Third element: \(-1400 \cdot 200 + 700 \cdot 75 = -280000 + 52500 = -227500\)- Fourth element: \(-1400 \cdot 300 + 700 \cdot 725 = -420000 + 507500 = 87500\)The result of \(B^{2} \cdot A^{2}\) is \[\begin{bmatrix} 332500 & 927500 \ -227500 & 87500 \end{bmatrix}\].

Key Concepts

Matrix Squaring

Matrix squaring is an important concept in matrix algebra where a matrix is multiplied by itself. It is denoted as \(A^{2} = A \cdot A\) or \(B^{2} = B \cdot B\). This operation is only possible when the number of columns in the first matrix matches the number of rows in the second matrix, which is always true for square matrices. To square a matrix, such as a 2x2 matrix A:

• First, create a new matrix where each element is calculated by performing the dot product of rows of the first matrix with columns of the second matrix.
• Repeat this process for all elements of the new 2x2 matrix.

Matrix squaring results in another matrix of the same dimensions, which can reveal intricate details about the original matrix's properties. When squaring 2x2 matrices, it's crucial to carefully calculate each element to avoid errors and ensure accuracy.

Matrix Algebra

Matrix algebra is a branch of mathematics that focuses on operations involving matrices, which are arrays of numbers arranged in rows and columns. These operations follow specific rules that allow for various manipulations and calculations. Common operations in matrix algebra include addition, subtraction, multiplication, and finding inverses. When working with matrices:

• Matrix addition and subtraction require the matrices to have the same dimensions.
• Matrix multiplication is only possible when the number of columns in the first matrix equals the number of rows in the second.

Each of these operations has its rules and properties, which are vital for solving linear equations and transforming geometric data. Mastery of these operations provides a foundation for understanding more complex mathematical concepts and real-world applications in fields like engineering and computer science.

2x2 Matrices

2x2 matrices are among the simplest types of matrices, consisting of two rows and two columns with elements typically denoted as \(a_{11}, a_{12}, a_{21}, a_{22}\). Their simplicity makes them a great tool for introducing matrix operations and demonstrating fundamental concepts of linear algebra.To perform operations on 2x2 matrices:

• The determinant can be calculated using the formula \(det(A) = a_{11}a_{22} - a_{12}a_{21}\).
• Inverses, if they exist, can be found by rearranging elements and dividing by the determinant.

2x2 matrices are often used in exercises to illustrate basic matrix multiplication and transformation principles, making them an essential building block in the study of matrices. Understanding 2x2 matrices helps build a foundation for studying more complex matrix forms and their applications.