Problem 44 - Precalculus Mathematics for Calculus

Question

(a) Prove that if \(A\) and \(B\) are \(2 \times 2\) matrices, then $$ (A+B)^{2}=A^{2}+A B+B A+B^{2} $$ (b) If \(A\) and \(B\) are \(2 \times 2\) matrices, is it necessarily true that $$ (A+B)^{2} \triangleq A^{2}+2 A B+B^{2} $$

Step-by-Step Solution

Answer

(a) Proven by expansion; (b) Not true, since \(AB \neq BA\) for matrices.

Step 1: Expand \((A + B)^2\)

We start by expanding the expression \((A + B)^2\) using the distributive property: \((A + B)(A + B) = A(A + B) + B(A + B)\).

Step 2: Use the Distributive Property

Next, apply the distributive property to each term: \[A(A + B) = A^2 + AB\] and \[B(A + B) = BA + B^2\].

Step 3: Combine All the Results

By combining the results from the previous steps, we obtain \( (A + B)(A + B) = A^2 + AB + BA + B^2 \), which is the desired expression.

Step 4: Compare \((A + B)^2\) with \(A^2 + 2AB + B^2\)

Compare the expanded expression for \((A + B)^2\) with \(A^2 + 2AB + B^2\). The key difference is the order of multiplication in the cross terms.

Step 5: Understand Matrix Multiplication

In matrix multiplication, \(AB\) and \(BA\) are not necessarily equal. Thus, \(A^2 + AB + BA + B^2\) is generally different from \(A^2 + 2AB + B^2\).

Step 6: Conclusion for Statement (b)

Since \(AB eq BA\) generally, \((A + B)^2\) is not equal to \(A^2 + 2AB + B^2\) for matrices.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra used to combine two matrices to produce another matrix. Unlike regular number multiplication, matrix multiplication involves specific rules on how to multiply entries. If you have two matrices, say matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( n \times p \), the product \( AB \) will have dimensions \( m \times p \).

When performing matrix multiplication, remember the following crucial points:

• The number of columns in the first matrix (\( A \)) must be the same as the number of rows in the second matrix (\( B \)).
• The element in the \( i \)-th row and \( j \)-th column of the resulting matrix is calculated by taking the dot product of the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).

This operation is not commutative, meaning \( AB eq BA \) in most cases.
Understanding that \( AB \) can be different from \( BA \) helps explain why the expressions \( (A + B)^2 = A^2 + AB + BA + B^2 \) and \( A^2 + 2AB + B^2 \) are different. The lack of commutativity is key here, and so we must account for it in calculations.

Distributive Property

The distributive property is a core concept that makes working with matrix and algebraic expressions simpler. It allows you to multiply a combined matrix expression efficiently. It states that for any three matrices \( A \), \( B \), and \( C \), whenever the operations are defined, you have:

• \( A(B + C) = AB + AC \)
• \((A + B)C = AC + BC \)

Matrix multiplication being applied through the distributive property can help us expand expressions like \((A + B)^2\).

When you apply this property to matrix products, you expand expressions by multiplying each matrix 'piece' by every other piece in the expression. This is exactly what we used when expanding \((A + B)(A + B)\) into \( A(A + B) + B(A + B) \) which simplifies further as \( A^2 + AB + BA + B^2 \).

This property shows clearly that care must be taken when working with non-commutative objects such as matrices, ensuring each step follows the order necessary for correctness.

Matrix Addition

Matrix addition is an essential element of matrix algebra that follows straightforward rules, similar to regular number addition. When adding two matrices, they must be of the same dimensions, meaning they must have the same number of rows and columns. Each entry in the resulting matrix is simply the sum of the corresponding entries from the original two matrices.

Think of matrices \( A \) and \( B \) as grids, where you add the number in each corresponding cell:

• For example, if \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \),
  then \( A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \).

This simplicity in matrix addition contrasts with the more complex matrix multiplication. It also underpins the straightforward representation of how to recombine terms in expanded matrix expressions. Understanding both will provide a more comprehensive grasp of matrix algebra and help in such proofs as \((A + B)^2 = A^2 + AB + BA + B^2\), where matrix addition combines the individual terms after distribution.