Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and \(\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].\) Show that \((A+B)(A-B) \neq A^{2}-B^{2}\)

Step-by-Step Solution

Answer

The calculation shows that \( (A + B)(A - B) = \left[\begin{array}{rr}3 & -2 \ 4 & 3\end{array}\right]\) and \( A^{2}-B^{2} = \left[\begin{array}{rr}4 & -5 \ 5 & 4\end{array}\right] \). Therefore, \( (A + B)(A - B) \neq A^{2}-B^{2} \).

Step 1: Calculate A + B

To calculate A + B, simply add together the corresponding elements in each matrix. That gives \[ A + B = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] + \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 1 & 1\end{array}\right]\].

Step 2: Calculate A - B

To calculate A - B, subtract the corresponding elements in each matrix. That gives \[ A - B = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] - \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}3 & -2 \ 1 & 5\end{array}\right]\].

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

To calculate the multiplication of these two matrices (A + B) and (A - B), multiply each element in the rows of the first matrix by each element in the columns of the second matrix and add them up which gives us \[ (A + B)(A - B) = \left[\begin{array}{rr}1 & 0 \ 1 & 1\end{array}\right]\cdot\left[\begin{array}{rr}3 & -2 \ 1 & 5\end{array}\right] = \left[\begin{array}{rr}3 & -2 \ 4 & 3\end{array}\right]\].

Step 4: Calculate A^2 and B^2

Calculate the square of A and B by multiplying each matrix by itself that results in \[ A^{2} = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right]\cdot\left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] = \left[\begin{array}{rr}3 & -6 \ 5 & 8\end{array}\right] \] and \[ B^{2} = \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right]\cdot\left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}-1 & -1 \ 0 & 4\end{array}\right]\].

Step 5: Calculate A^2 - B^2

Subtract B^2 from A^2 to get \[ A^{2}-B^{2} = \left[\begin{array}{rr}3 & -6 \ 5 & 8\end{array}\right]- \left[\begin{array}{rr}-1 & -1 \ 0 & 4\end{array}\right] = \left[\begin{array}{rr}4 & -5 \ 5 & 4\end{array}\right]\]

Key Concepts

Matrix Addition

Matrix addition involves adding each element of one matrix to the corresponding element of another matrix. The matrices must be of the same dimensions. For example, if you have matrix \[A = \begin{bmatrix} 2 & -1 \ 1 & 3 \end{bmatrix}\] and \[B = \begin{bmatrix} -1 & 1 \ 0 & -2 \end{bmatrix},\]their sum is calculated by adding each pair of corresponding elements:

• Add the top-left elements: 2 + (-1) = 1
• Add the top-right elements: -1 + 1 = 0
• Add the bottom-left elements: 1 + 0 = 1
• Add the bottom-right elements: 3 + (-2) = 1

Thus, the resulting matrix is:\[A + B = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}.\]Adding matrices is straightforward when they have the same size. Remember this rule: add element-by-element.

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead of adding, you subtract corresponding elements from each other. If we continue with matrices A and B:\[A = \begin{bmatrix} 2 & -1 \ 1 & 3 \end{bmatrix}\] and \[B = \begin{bmatrix} -1 & 1 \ 0 & -2 \end{bmatrix},\]then subtracting B from A involves:

• Subtracting the top-left elements: 2 - (-1) = 3
• Subtracting the top-right elements: -1 - 1 = -2
• Subtracting the bottom-left elements: 1 - 0 = 1
• Subtracting the bottom-right elements: 3 - (-2) = 5

Thus, the resulting matrix is:\[A - B = \begin{bmatrix} 3 & -2 \ 1 & 5 \end{bmatrix}.\]When performing subtraction, ensure you handle negative numbers carefully; the sign changes matter.

Matrix Multiplication

Matrix multiplication is more complex than addition and subtraction. To multiply two matrices, where the number of columns in the first matrix is equal to the number of rows in the second matrix, multiply each row element of the first matrix by the corresponding column element of the second matrix and sum the products.For matrices \[(A + B) = \begin{bmatrix} 1 & 0 \ 1 & 1 \end{bmatrix}\] and \[(A - B) = \begin{bmatrix} 3 & -2 \ 1 & 5 \end{bmatrix},\]we calculate:

• For the top-left corner: \((1 \times 3) + (0 \times 1) = 3\)
• For the top-right corner: \((1 \times -2) + (0 \times 5) = -2\)
• For the bottom-left corner: \((1 \times 3) + (1 \times 1) = 4\)
• For the bottom-right corner: \((1 \times -2) + (1 \times 5) = 3\)

Resulting in:\[(A + B)(A - B) = \begin{bmatrix} 3 & -2 \ 4 & 3 \end{bmatrix}.\]Practicing this will improve your confidence in seeing the patterns.

Matrix Squaring

Matrix squaring means multiplying a matrix by itself. Each element is calculated as in matrix multiplication, applied back to the same matrix.For \[A = \begin{bmatrix} 2 & -1 \ 1 & 3 \end{bmatrix},\] square it by multiplying it with itself:

• Top-left: \((2 \times 2) + (-1 \times 1) = 3\)
• Top-right: \((2 \times -1) + (-1 \times 3) = -6\)
• Bottom-left: \((1 \times 2) + (3 \times 1) = 5\)
• Bottom-right: \((1 \times -1) + (3 \times 3) = 8\)

Giving:\[A^{2} = \begin{bmatrix} 3 & -6 \ 5 & 8 \end{bmatrix}.\]Repeat for B and remember each matrix must have matching rows and columns.

Properties of Matrices

There are several properties essential for understanding matrix operations:

• Commutative Property: Does not hold for matrix multiplication. That means, in general, \(AB eq BA\).
• Associative Property: Holds for both addition and multiplication, so \((AB)C = A(BC)\).
• Distributive Property: Applies over addition and subtraction, such as \(A(B + C) = AB + AC\).
• Identities: The identity matrix, when multiplied with another matrix, leaves it unchanged: \(AI = IA = A\).

Understanding these properties helps in solving complex matrix problems efficiently and verifying solutions are accurate.