Problem 103 - 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].$$$\text { Show that }(A+B)^{2} \neq A^{2}+2 A B+B^{2}$$

Step-by-Step Solution

Answer

The given statement (A+B)^{2} = A^{2}+2 A B+B^{2} is not correct for the provided matrices A and B

Step 1: Compute A + B

First, start by adding up matrices A and B. To do this, just add up the corresponding entries from both matrices. The result should be: \[ A + B= \left[\begin{array}{rr}2-1 & -1+1 \ 1+0 & 3-2\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 1 & 1\end{array}\right] \]

Step 2: Compute (A+B)^{2}

Next, square up the resulting matrix from the previous step. To do this, multiply the matrix with itself. \[ (A+B)^2 = \left[\begin{array}{rr}1 & 0 \ 1 & 1\end{array}\right] \times \left[\begin{array}{rr}1 & 0 \ 1 & 1\end{array}\right] = \left[\begin{array}{rr}1 & 0 \ 2 & 1\end{array}\right] \]

Step 3: Compute A^2, 2AB and B^2

Now, find out the square of matrix A, the product of 2AB, and the square of matrix B. \[ A^2 = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] \times \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] = \left[\begin{array}{rr}3 & -7 \ 5 & 8\end{array}\right] \] \[ 2AB = 2 \times \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] \times \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}-4 & 4 \ -2 & 0\end{array}\right] \] \[ B^2 = \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] \times \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}-1 & 0 \ 0 & 4\end{array}\right] \]

Step 4: Compute A^2 + 2AB + B^2

The fourth step is to add the matrices obtained from the previous step, i.e., A^2, 2AB and B^2. \[ A^2 + 2AB + B^2 = \left[\begin{array}{rr}3 & -7 \ 5 & 8\end{array}\right] + \left[\begin{array}{rr}-4 & 4 \ -2 & 0\end{array}\right] + \left[\begin{array}{rr}-1 & 0 \ 0 & 4\end{array}\right] = \left[\begin{array}{rr}-2 & -3 \ 3 & 12\end{array}\right] \]

Step 5: Comparison

Now compare the resulting matrices from Step 2: (A+B)^{2} and Step 4: A^{2}+2 A B+B^{2}. They are different, therefore the given statement, (A+B)^{2} = A^{2}+2 A B+B^{2}, is not true for given matrices A and B.

Key Concepts

Matrices Addition

Matrix addition is a fundamental operation in matrix algebra, symbolizing the combination of two matrices of the same size. Matrices are added by combining corresponding elements from each matrix. For instance, when we add two matrices, say, Matrix A and Matrix B, the resulting matrix's element in the ith row and jth column (C_ij) is given by the sum of the elements A_ij and B_ij from Matrix A and Matrix B respectively.

It's crucial to note that this operation is only possible with matrices of the same dimensions. For example, consider matrices A and B from the textbook exercise. Their addition is straightforward:

• Add the corresponding entries of both matrices.
• Ensure both matrices are of the same size.

After following these steps, we get a new matrix, each of whose elements is the sum of the corresponding elements from A and B. This adheres to the commutative property of matrix addition, meaning that A + B is the same as B + A as demonstrated in the solution.

Matrix Multiplication

Matrix multiplication is slightly more complex than matrix addition and entails a specific rule set. When multiplying Matrix A by Matrix B (A × B), the element in the resulting matrix's ith row and jth column is the dot product of the elements in A's ith row and B's jth column.

To compute the product, each element of the ith row of A is multiplied with the corresponding element of the jth column of B, and the results are summed up to produce a single element in the resulting matrix. Remember:

• The number of columns in A must equal the number of rows in B.
• The resulting matrix will have the same number of rows as A and the same number of columns as B.

In the given example, we multiplied A with B as well as A + B with itself. The different outcomes show that matrix multiplication is not commutative (AB ≠ BA in general) which is an essential property to remember.

Matrix Squaring

Matrix squaring is a specific case of matrix multiplication in which a matrix is multiplied by itself. This can be denoted as A² for matrix A. The textbook's exercise provides an example where matrix A + B is squared. This action entails multiplying the matrix by itself, applying the rules of matrix multiplication as practiced in the previous section.

While squaring, it's important to remember that:

• The matrix needs to be square — that is, having the same number of rows and columns.
• Even though the matrix is being multiplied by itself, we still perform regular matrix multiplication row-by-column.

As shown in the solution, squaring a matrix doesn't necessarily lead to simply squaring each individual element. Instead, the entire matrix multiplication process must be followed.

Properties of Matrix Operations

The properties of matrix operations set the ground rules for how matrices can be manipulated. Understanding these properties is crucial in performing algebraic manipulations with matrices. The exercise highlights that some properties familiar from regular algebra do not necessarily hold in matrix algebra. For example, the distributive property does appear to apply when expanding the binomial ((A + B)²), but the resulting matrices reveal a contradiction.

Some essential properties include:

• Commutative Property for Addition: A + B = B + A
• Associative Property: (A + B) + C = A + (B + C) for addition and (AB)C = A(BC) for multiplication.
• Distributive Property: A(B + C) = AB + AC and (A + B)C = AC + BC.

However, the non-commutative nature of multiplication (AB ≠ BA) and the counterexample provided by squaring (A + B) demonstrate the nuanced differences in matrix algebra compared to scalar algebra. These properties are vital for students to comprehend, as they form the basis of more advanced matrix operations and applications.