Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Let \(A=\left[\begin{array}{rr}1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{rr}2 & -1 \\ 3 & 1\end{array}\right], \quad C=\left[\begin{array}{rr}3 & 1 \\ -2 & 0\end{array}\right]\) $$ (A+B)(A-B) \neq A^{2}-B^{2}, \text { where } A^{2}=A A \text { and } B^{2}=B B $$

Step-by-Step Solution

Answer

The matrices (A + B)(A - B) and A^2 - B^2 are not equal due to cross terms in the expansion of (A + B)(A - B), unlike the definition of A^2 - B^2. This results in different matrices.

Step 1: Calculate A + B

To find \(A + B\), add corresponding elements from matrices \(A\) and \(B\):\[A + B = \begin{bmatrix} 1+2 & 2+(-1) \ 0+3 & -3+1 \end{bmatrix} = \begin{bmatrix} 3 & 1 \ 3 & -2 \end{bmatrix}\]

Step 2: Calculate A - B

To find \(A - B\), subtract corresponding elements of \(B\) from \(A\):\[A - B = \begin{bmatrix} 1-2 & 2-(-1) \ 0-3 & -3-1 \end{bmatrix} = \begin{bmatrix} -1 & 3 \ -3 & -4 \end{bmatrix}\]

Step 3: Calculate (A + B)(A - B)

Multiply the matrices \( (A + B) \) and \( (A - B) \):\[(A + B)(A - B) = \begin{bmatrix} 3 & 1 \ 3 & -2 \end{bmatrix}\begin{bmatrix} -1 & 3 \ -3 & -4 \end{bmatrix}\]

Key Concepts

Matrices

Matrices are a fundamental part of linear algebra, used to organize numerical information in a structured form. A matrix is essentially a rectangular array of numbers arranged in rows and columns. In the context of mathematics and especially in operations involving matrices, the term matrix refers to collections like these where operations such as addition, subtraction, and multiplication are conducted.

Each specific number in a matrix is known as an element. The size or dimension of a matrix is defined by the number of rows and columns it has. For example, the matrix \[A=\left[\begin{array}{rr}1 & 2 \ 0 & -3\end{array}\right]\] is a 2x2 matrix, meaning it has 2 rows and 2 columns.

Understanding how to manipulate these arrays paves the way for exploring more advanced topics in mathematics, physics, engineering, and computer science. Matrices can represent systems of equations, transform geometric entities, and are vital in representing data in machine learning algorithms.

Matrix Addition

Matrix addition, like you see in the solution step, involves adding two matrices together by adding their corresponding elements. For matrices to be added, they must be of the same dimension. This means every element from one matrix should have a corresponding element in the other.

• Dimensions: Both matrices need to have the same number of rows and columns.
• Element Addition: Add the numbers located in the same position in each matrix.

As demonstrated in the solution, adding matrices A and B would involve operations like \(1 + 2\) and \(2 + (-1)\), constructing a new matrix from these summarized values:
\[A + B = \begin{bmatrix} 3 & 1 \ 3 & -2 \end{bmatrix}\]
This process is straightforward as long as dimensions are matched, ensuring a seamless element-by-element addition.

Matrix Subtraction

Matrix subtraction follows a similar principle to matrix addition, yet the process involves subtracting corresponding elements. Just as with addition, the matrices involved must share the same dimensions.

• Dimensions: Ensure both matrices involved in the subtraction are of the same size.
• Element Subtraction: Subtract the element in the same position of the second matrix from that in the first matrix.

Using the matrices A and B from the exercise, elements are subtracted as follows: \(1 - 2\) and \(2 - (-1)\), forming a new matrix composed of these differences:
\[A - B = \begin{bmatrix} -1 & 3 \ -3 & -4 \end{bmatrix}\]
This method requires careful attention to detail to ensure accurate subtraction, as any small error can lead to a completely incorrect matrix outcome.