Problem 48 - Precalculus: Mathematical 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 $$(A+B)^{2} \stackrel{?}{=} A^{2}+2 A B+B^{2}$$

Step-by-Step Solution

Answer

(a) True; (b) False, unless A and B commute.

Step 1: Expand the Square of Sum

To prove part (a), start by expanding \((A+B)^2\). The expression is defined as the matrix multiplication of \(A+B\) with itself, i.e., \((A+B) \cdot (A+B)\). Applying the distributive property of matrix multiplication, \[(A+B)(A+B) = A(A+B) + B(A+B) \]\(=AA + AB + BA + BB = A^2 + AB + BA + B^2\).This expansion matches the right side of the equation \(A^2 + AB + BA + B^2\). Thus, part (a) is proved.

Step 2: Verify Part (b)

For part (b), we need to determine if the equation \((A+B)^2 = A^2 + 2AB + B^2\) is always true for any 2x2 matrices \(A\) and \(B\).Consider what would result from assuming the middle term of this equation is incorrect. Recall from part (a) that \[(A+B)^2 = A^2 + AB + BA + B^2\].This doesn't simplify to \((A+B)^2 = A^2 + 2AB + B^2\) unless \(AB = BA\), which is only true if \(A\) and \(B\) commute. Therefore, without the commutative property (\(AB = BA\)), the middle terms \(AB\) and \(BA\) are not necessarily equal, making part (b) false in general.

Step 3: Conclusion

Finally, summarize the findings. In part (a), we established that for any 2x2 matrices \(A\) and \(B\), \((A+B)^2 = A^2 + AB + BA + B^2\). In part (b), we demonstrated that \((A+B)^2 = A^2 + 2AB + B^2\) is only true when \(AB = BA\), or in other words, when \(A\) and \(B\) commute.

Key Concepts

Understanding 2x2 Matrices

Matrices are a fundamental concept in linear algebra. A matrix is essentially a rectangular array of numbers.
In this case, we're focusing on 2x2 matrices, which have two rows and two columns. The general form of a 2x2 matrix is displayed as follows: \[\begin{bmatrix}a & b \c & d \end{bmatrix}\] Here, \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix. Matrix operations, including multiplication, are staple processes performed on these constructs. When performing any operation on matrices, remember the order of input elements, as it is crucial for the accuracy of results.
This is because matrix multiplication is not commutative in general, which means that \(AB\) is not necessarily equal to \(BA\).

Exploring Matrix Algebra

Matrix algebra involves various operations such as addition, subtraction, and multiplication. Each operation has its own rules and properties. To perform addition or subtraction between two matrices, they must be of the same size.
So, two 2x2 matrices can be added or subtracted if they are both 2x2 matrices. Simply perform the operations element-wise:

• Addition: \(\begin{bmatrix}a & b\c & d\end{bmatrix} + \begin{bmatrix}e & f\g & h\end{bmatrix} = \begin{bmatrix}a+e & b+f\c+g & d+h\end{bmatrix}\)
• Subtraction is similar, with subtraction performed element-wise.

Matrix multiplication is more complex. To multiply two 2x2 matrices \(A\) and \(B\), you calculate the dot products of rows from \(A\) with columns from \(B\). This operation can lead to different results based on the order, highlighting the non-commutative nature of matrix operations.

The Distributive Property of Matrices

Matrix operations, particularly multiplication, share several important properties similar to ordinary arithmetic. One such property is the distributive property, which is crucial when expanding expressions like \((A+B)^2\).
For matrices \(A\) and \(B\), the distributive property allows for the following:
\[A(B+C) = AB + AC\]This means multiplication is distributed over addition, facilitating the expansion of expressions. When expanding \((A+B)(A+B)\), the distributive property reveals the interaction between terms:

• \(A(A+B) = A^2 + AB\)
• \(B(A+B) = BA + B^2\)

Resulting in:\[A^2 + AB + BA + B^2\]Notice how this differs from regular algebra due to the presence of both \(AB\) and \(BA\), highlighting again that matrices generally do not commute. Understanding how these properties transfer to matrices helps solve problems effectively and accurately.