Problem 48 - Algebra and Trigonometry

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 $$ (A+B)^{2} \stackrel{3}{\perp} A^{2}+2 A B+B^{2} $$

Step-by-Step Solution

Answer

(a) Yes, proven. (b) No, not necessarily.

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

Let's begin by expanding the square of the sum of two matrices. We have \[(A+B)^{2} = (A+B)(A+B)\]. Using the distributive law for matrix multiplication, this is equal to \[A(A+B) + B(A+B)\]. Continue expanding: \[= AA + AB + BA + BB\]. Since \(A^2 = AA\) and \(B^2 = BB\), we finally get \[= A^2 + AB + BA + B^2\].

Step 2: Understand the given expression

The expression provided to prove is \[A^2 + AB + BA + B^2\]. The steps in the expansion match this expression, confirming the identity. Thus, we've shown that \((A+B)^{2} = A^{2} + AB + BA + B^2\) as required.

Step 3: Analyze the second statement

Let's analyze the statement \((A+B)^{2} \overset{?}{=} A^2 + 2AB + B^2\). Recall from Step 1 that \((A+B)^{2} = A^2 + AB + BA + B^2\). Notice the difference is in the middle terms: \(AB + BA\) in contrast to \(2AB\).

Step 4: Check matrix multiplication properties

Note that matrix multiplication is not commutative, meaning generally \(AB eq BA\). Therefore, \(AB + BA\) \(eq 2AB\) unless \(AB = BA\). However, without additional information about \(A\) and \(B\), we cannot assume that \(AB = BA\). Hence, \((A+B)^{2} eq A^2 + 2AB + B^2\) unless \(A\) and \(B\) commute.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in matrix algebra, and it involves combining two matrices to produce a new matrix. The process might sound simple, but it involves a specific rule and is not as straightforward as multiplying numbers. In matrix multiplication, the element in the `i-th` row and `j-th` column of the resulting matrix is obtained by taking the dot product of the `i-th` row of the first matrix with the `j-th` column of the second matrix.

It's crucial to remember that matrices must have compatible dimensions to be multiplied. Typically, if matrix \(A\) is of size \(m \times n\) and matrix \(B\) is of size \(n \times p\), then their product \(AB\) will have dimensions \(m \times p\).

• Matrix multiplication is associative but not always commutative.
  
• The order of multiplication matters—\(AB\) is not necessarily the same as \(BA\).
  

This unique property plays a significant role in understanding the nature of the expression expansions in the original exercise.

Commutative Property

The commutative property is a fundamental concept in mathematics, usually applying to operations like addition and multiplication of numbers, allowing changes in the order of the operands without affecting the result. However, when it comes to matrix multiplication, this property generally does not apply. In other words, for matrices \(A\) and \(B\), the equation \(AB = BA\) does not hold in most cases.

In the context of matrices, whether \(AB = BA\) depends on the specific matrices you're dealing with. For the general case:

• Non-commutative nature: This means order matters: \(AB eq BA\).
  
• Equal products: Only if \(AB = BA\), the matrices are said to commute.
  

In our exercise, when proving \((A+B)^2 = A^2 + AB + BA + B^2\), the commutative property is crucial to note. \(AB + BA\) only equals \(2AB\) if \(A\) and \(B\) commute, which is not usually the case without additional information about \(A\) and \(B\). Understanding this property helps in correctly expanding and simplifying matrix expressions.

Distributive Law

The distributive law is an essential principle in algebra that also applies to matrix operations. It indicates that multiplying a matrix by a sum is the same as multiplying the matrix by each element of the sum separately and then adding the results. For matrices \(A\), \(B\), and \(C\), the distributive property is expressed as:

• \(A(B + C) = AB + AC\)
  
• \((B + C)A = BA + CA\)
  

This property is utilized in our original exercise when expanding \((A+B)^2\). By applying the distributive law, we derive:
\[ (A + B)(A + B) = A(A + B) + B(A + B) = A^2 + AB + BA + B^2 \]
Each step demonstrates the power of the distributive law in simplifying expressions involving matrices. This law allows matrices to be multiplied through each individual component before summing the results, providing a clear path to managing complex matrix operations. With this understanding, students can solve similar matrix expression problems confidently.