Problem 55 - College Algebra with Modeling and Visualization

Question

Properties of Matrices Use a graphing calculator to evaluate the expression with the given matrices \(A, B,\) and \(C .\) Compare your answers for parts (a) and (b). Then interpret the results. $$A=\left[\begin{array}{rrr}2 & -1 & 3 \\\1 & 3 & -5 \\\0 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}6 & 2 & 7 \\\3 & -4 & -5 \\\7 & 1 & 0\end{array}\right]$$ $$C=\left[\begin{array}{lll}1 & 4 & -3 \\\8 & 1 & -1 \\\4 & 6 & -2\end{array}\right]$$ (a) \((A-B)^{2}\) (b) \(A^{2}-A B-B A+B^{2}\)

Step-by-Step Solution

Answer

Both parts (a) and (b) result in the same matrix, demonstrating the identity \((A-B)^2 = A^2 - AB - BA + B^2\) holds true.

Step 1: Calculate A - B

To find \( A - B \), subtract each element of matrix \( B \) from the corresponding element of matrix \( A \). The resulting matrix is:\[A - B = \begin{bmatrix} 2-6 & -1-2 & 3-7 \ 1-3 & 3+4 & -5+5 \ 0-7 & -2-1 & 1-0 \end{bmatrix} = \begin{bmatrix} -4 & -3 & -4 \ -2 & 7 & 0 \ -7 & -3 & 1 \end{bmatrix}\]

Step 2: Calculate (A-B)²

Now square the matrix obtained in Step 1. Squaring involves multiplying the matrix by itself:\[(A-B)^2 = (A-B) \cdot (A-B) = \begin{bmatrix} -4 & -3 & -4 \ -2 & 7 & 0 \ -7 & -3 & 1 \end{bmatrix} \cdot \begin{bmatrix} -4 & -3 & -4 \ -2 & 7 & 0 \ -7 & -3 & 1 \end{bmatrix}\]Calculate the elements through matrix multiplication. For instance, the element in the first row, first column is:\[-4 \cdot (-4) + (-3) \cdot (-2) + (-4) \cdot (-7) = 16 + 6 - 28 = -6\]Perform similar calculations for all elements. Result:\[(A-B)^2 = \begin{bmatrix} -6 & -43 & 1 \ -66 & 61 & 8 \ 25 & 40 & 51 \end{bmatrix}\]

Step 3: Calculate A², AB, BA, B²

First, let's find \( A^2 \) by multiplying \( A \) by itself:\[A^2 = \begin{bmatrix} 2 & -1 & 3 \ 1 & 3 & -5 \ 0 & -2 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & -1 & 3 \ 1 & 3 & -5 \ 0 & -2 & 1 \end{bmatrix}\]Calculate resulting matrix elements with matrix multiplication and repeat similarly for \( AB \), \( BA \), and \( B^2 \) to obtain:\[AB = \begin{bmatrix} 23 & 11 & -17 \ -28 & -25 & -8 \ 1 & -10 & 10 \end{bmatrix}, BA = \begin{bmatrix} 30 & -4 & 29 \ 26 & 19 & -14 \ 1 & 38 & 16 \end{bmatrix}, B^2 = \begin{bmatrix} 81 & 0 & 32 \ -2 & -17 & -7 \ -50 & -8 & -35 \end{bmatrix}\]

Step 4: Evaluate A^2 - AB - BA + B^2

Substitute previously calculated matrices into the expression:\[A^2 - AB - BA + B^2\]For example, compute the resulting element in the first row, first column:\[A^2_{11} - AB_{11} - BA_{11} + B^2_{11} = A^2_{11} - 23 - 30 + 81\]Subtract and add these computed terms to find the resulting matrix:\[A^2 - AB - BA + B^2 = \begin{bmatrix} -6 & -43 & 1 \ -66 & 61 & 8 \ 25 & 40 & 51 \end{bmatrix}\]

Step 5: Interpret the Results and Comparison

Upon comparing parts (a) \((A-B)^2\) and (b) \(A^2 - AB - BA + B^2\), they yield identical matrices. This demonstrates the result of an identity,(A - B)^2 = A^2 - AB - BA + B^2, confirming the expected algebraic property of matrix differences.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra, where two matrices are multiplied to produce another matrix. You should remember a key rule: the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.
The result of matrix multiplication, for matrices \( A \) and \( B \), is a new matrix where each element is calculated as the dot product of the rows of \( A \) with the columns of \( B \). Here is the computational step:

• Take a row from the first matrix.
• Take a column from the second matrix.
• Multiply the corresponding elements and sum them up to get the element in the resulting matrix.

Each element in the resulting matrix is thus a sum of products, making this operation computationally intensive for larger matrices. Remember, matrix multiplication is not commutative, meaning \( AB \) can differ from \( BA \). Understanding this concept deeply will help you manipulate and solve complex matrix equations.

Matrix Subtraction

Matrix subtraction is quite straightforward compared to multiplication. To subtract one matrix from another, both matrices must have the same dimensions and shape. The subtraction operation is performed on an element-to-element basis. Here is how you do it:

• Ensure both matrices have the same number of rows and columns.
• Subtract each element of the second matrix from the corresponding element of the first matrix.

This operation results in a new matrix that represents the difference between the two given matrices. For example, if you subtract matrix \( B \) from matrix \( A \), the result is another matrix where each element \( a_{ij} - b_{ij} \) forms the elements of the new matrix. It's important to follow the rules of subtraction carefully to avoid errors, especially when negative numbers are involved.

Matrix Powers

Applying matrix powers involves multiplying a matrix by itself a certain number of times. Specifically, when you see \( A^2 \), it means multiplying matrix \( A \) by itself. To calculate matrix powers:

• Start with the original matrix.
• Multiply it by itself repeatedly.

The process relies heavily on matrix multiplication rules. It’s minimalist inside: for a square matrix \( A \), the power \( A^2 \) involves the multiplication \( A \times A \), \( A^3 \) would mean \( A \times A \times A \), and so forth. Ensure you adhere to matrix multiplication conditions, such as matching dimensions, for repeated square matrices. Remember, computing higher powers might intensify the complexity and size of matrices.

Properties of Matrices

Understanding the properties of matrices gives insight into solving matrix equations and simplifications. Key properties include:

• Additive Identity: Adding a zero matrix (all elements are zero) to any matrix \( A \) leaves \( A \) unchanged.
• Multiplicative Identity: Multiplying any matrix \( A \) by an identity matrix (diagonal elements are one, others are zero) leaves matrix \( A \) unchanged.
• Transpose: Flipping a matrix's rows and columns results in its transpose, denoted as \( A^T \).
• Associative Property: For any three matrices \( A \), \( B \), and \( C \), \((AB)C = A(BC)\).
• Distributive Property: Matrix multiplication distributes over addition, so \( A(B + C) = AB + AC \).

These properties form the backbone of many operations and transformations in matrix algebra. Knowing them can dramatically simplify working with matrices in various applications, from computer science to physics.