Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \((A-B)^{2}\) (b) \(A^{2}-A B-B A+B^{2}\)

Step-by-Step Solution

Answer

The matrices resulting from \((A-B)^{2}\) and \(A^{2}-AB-BA+B^{2}\) are to be compared. If they are the same, then the statements in (a) and (b) are equivalent. If not, they are not the same.

Step 1: Calculate A-B

Subtract matrix B from A. To subtract the matrices, subtract corresponding elements. The new matrix, say D, is calculated as follows: D = A - B = \[\begin{bmatrix} 1-2 & 2-3 & -1-0 \ 0-4 & -2-1 & 3-(-2) \ 4-(-1) & -3-2 & 2-0 \end{bmatrix}\] = \[\begin{bmatrix} -1 & -1 & -1 \ -4 & -3 & 5 \ 5 & -5 & 2 \end{bmatrix}\].

Step 2: Calculate \((A-B)^{2}\)

Calculate the square of the new matrix D, i.e., perform matrix multiplication D*D:D*D = \(\[\begin{bmatrix}-1 & -1 & -1 \-4 & -3 & 5 \5 & -5 & 2\end{bmatrix}\]\) * \(\[\begin{bmatrix}-1 & -1 & -1 \-4 & -3 & 5 \5 & -5 & 2\end{bmatrix}\]\)This operarion results in a new matrix E.

Step 3: Calculate matrix products for part (b)

In order to evaluate \(A^{2}-AB-BA+B^{2}\), compute the following products: \(A*A, A*B, B*A, B*B\). The results of these computations will be four new matrices.

Step 4: Solve for \(A^{2}-AB-BA+B^{2}\)

Now, compute \(A^{2}-AB-BA+B^{2}\) by subtracting and adding the matrices obtained in the previous step. The result will be another new matrix.

Step 5: Compare Results

Finally, compare the two result matrices by inspecting if they are equal or not. If they are equal the statements in part (a) and part (b) are equivalent. If not, they are different.

Key Concepts

Subtracting Matrices

Matrix subtraction is a fundamental operation in linear algebra that involves taking two matrices of the same dimensions and subtracting the corresponding entries. This operation is performed element-by-element, resulting in a new matrix where each element represents the difference of elements in similar positions.

For instance, if we have two matrices, \( \text{Matrix } A = \begin{bmatrix}a_{11} & a_{12}\ a_{21} & a_{22}\end{bmatrix} \) and \( \text{Matrix } B = \begin{bmatrix}b_{11} & b_{12}\ b_{21} & b_{22}\end{bmatrix} \), their subtraction, \( A - B \), yields \( \begin{bmatrix}a_{11} - b_{11} & a_{12} - b_{12}\ a_{21} - b_{21} & a_{22} - b_{22}\end{bmatrix} \).

Matrix subtraction is used in various applications such as solving systems of linear equations, transforming geometric shapes, and analyzing differences in data.

Matrix Multiplication

Matrix multiplication is not as straightforward as addition or subtraction and involves a special set of rules. It is a binary operation that takes a pair of matrices, and produces another matrix. However, for the multiplication to be valid, the number of columns in the first matrix must equal the number of rows in the second matrix.

When multiplying two matrices, the element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the corresponding column from the second matrix. For example, when multiplying matrix \(A\) and matrix \(B\), the element on the \(i\)-th row and \(j\)-th column in the resulting matrix \(C\) is computed as \( c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{in}b_{nj} \), assuming matrix \(A\) has \(n\) columns.

The process involves multiplying each element of the row by the corresponding element of the column and then summing those products. This product is essential in various fields including computer graphics, economics, and social science, where relationships between datasets represented by matrices need to be established.

Matrix Square

The concept of squaring a matrix, denoted as \(A^2\), is analogous to squaring a number, except it involves matrix multiplication. The square of a matrix \(A\) is simply the product of \(A\) with itself, \(A \times A\). It's important to note that not all matrices can be squared; the matrix needs to be square itself (same number of rows and columns) to multiply it by itself, adhering to the rules of matrix multiplication.

In squaring a matrix, we multiply each row of \(A\) with each column of \(A\), which results in a new matrix where each element signifies the relationship of the rows and columns within the original matrix. This operation is used in higher-level mathematics, such as in the study of linear transformations and eigenvalues.

Graphing Utility Matrices

Graphing utility matrices refer to the use of computational tools to handle complex matrix operations. Graphing calculators and computer software are equipped with functions to perform matrix addition, subtraction, multiplication, and more advanced operations like finding determinants or inverses.

These tools are especially useful in visualizing computations that are too tedious to be carried out by hand, and in teaching and learning environments where they can help students understand the properties of matrices more intuitively. For example, a graphing utility can quickly demonstrate the effect of subtracting one matrix from another or the result of multiplying matrices, providing immediate visual feedback on the operation's outcome.

It is advisable to comprehend the manual process of matrix operations to gain a deeper understanding; subsequently, graphing utilities can be leveraged to verify complex calculations or to handle large matrices efficiently in practical applications.