Problem 31 - Precalculus

Question

Let $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{rr} 0 & -3 \\ -5 & 2 \end{array}\right] $$ Compare \((A+B)^{2}\) to \(A^{2}+2 A B+B^{2}\). Discuss with your classmates what constraints must be placed on two arbitrary matrices \(A\) and \(B\) so that both \((A+B)^{2}\) and \(A^{2}+2 A B+B^{2}\) exist. When will \((A+B)^{2}=A^{2}+2 A B+B^{2} ?\) In general, what is the correct formula for \((A+B)^{2} ?\)

Step-by-Step Solution

Answer

For square matrices, both expressions exist, but \((A+B)^2\) equals \(A^2 + 2AB + B^2\) only if matrices are commutative. Generally, \((A+B)^2 = A^2 + AB + BA + B^2\).

Step 1: Calculate A+B

Add matrices \(A\) and \(B\) element-wise. Each element of the resulting matrix is the sum of the corresponding elements from \(A\) and \(B\): \[A + B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 1+0 & 2-3 \ 3-5 & 4+2 \end{bmatrix} = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix}\]

Step 2: Compute (A+B)^2

Multiply the matrix \(A + B\) with itself to find \((A+B)^2\). The multiplication is performed by taking the dot product of rows and columns: \[(A+B)^2 = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix} \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix} = \begin{bmatrix} (1)(1) + (-1)(-2) & (1)(-1) + (-1)(6) \ (-2)(1) + (6)(-2) & (-2)(-1) + (6)(6) \end{bmatrix} = \begin{bmatrix} 3 & -7 \ -14 & 38 \end{bmatrix}\]

Step 3: Calculate A^2

Multiply matrix \(A\) with itself to find \(A^2\): \[A^2 = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} (1)(1) + (2)(3) & (1)(2) + (2)(4) \ (3)(1) + (4)(3) & (3)(2) + (4)(4) \end{bmatrix} = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix}\]

Step 4: Calculate 2AB

Multiply \(A\) and \(B\) and then multiply the result by 2. First, calculate \(AB\): \[AB = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} (1)(0) + (2)(-5) & (1)(-3) + (2)(2) \ (3)(0) + (4)(-5) & (3)(-3) + (4)(2) \end{bmatrix} = \begin{bmatrix} -10 & 1 \ -20 & -1 \end{bmatrix}\] Then multiply by 2: \[2AB = 2 \begin{bmatrix} -10 & 1 \ -20 & -1 \end{bmatrix} = \begin{bmatrix} -20 & 2 \ -40 & -2 \end{bmatrix}\]

Step 5: Calculate B^2

Multiply matrix \(B\) with itself: \[B^2 = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} (0)(0) + (-3)(-5) & (0)(-3) + (-3)(2) \ (-5)(0) + (2)(-5) & (-5)(-3) + (2)(2) \end{bmatrix} = \begin{bmatrix} 15 & -6 \ -10 & 19 \end{bmatrix}\]

Step 6: Sum A^2, 2AB, and B^2

Add \(A^2\), \(2AB\), and \(B^2\) together: \[A^2 + 2AB + B^2 = \begin{bmatrix} 7 & 10 \ 15 & 22 \end{bmatrix} + \begin{bmatrix} -20 & 2 \ -40 & -2 \end{bmatrix} + \begin{bmatrix} 15 & -6 \ -10 & 19 \end{bmatrix} = \begin{bmatrix} 2 & 6 \ -35 & 39 \end{bmatrix}\]

Step 7: Compare (A+B)^2 and A^2 + 2AB + B^2

Comparing matrices \((A+B)^2 = \begin{bmatrix} 3 & -7 \ -14 & 38 \end{bmatrix}\) and \(A^2 + 2AB + B^2 = \begin{bmatrix} 2 & 6 \ -35 & 39 \end{bmatrix}\), they are not equal. This shows that the expansion \((A+B)^2 = A^2 + 2AB + B^2\) does not hold in this case.

Step 8: Discuss Matrix Multiplication Constraints

Both \(A+B\) and \(A^2 + 2AB + B^2\) exist for square matrices of the same size only. For \((A+B)^2\) to equal \(A^2 + 2AB + B^2\), matrix multiplication properties (like commutativity) must hold, but multiplication of matrices is generally not commutative.

Step 9: General Formula for (A+B)^2

The general formula for \((A+B)^2\) is derived by simplifying the expression \((A+B)(A+B) = A^2 + AB + BA + B^2\). The formula simplifies to \(A^2 + 2AB + B^2\) only if \(AB = BA\), but this is not typically the case for matrices.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation that involves combining corresponding elements of two matrices. Given two matrices of the same dimensions, the result of their addition is another matrix where each element is the sum of the corresponding elements from the original matrices.

For example, if you have matrices \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \), adding them together is straightforward:

• Add element-wise: \( A + B = \begin{bmatrix} 1+0 & 2-3 \ 3-5 & 4+2 \end{bmatrix} = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix} \).

Thus, matrix addition requires that both matrices have the same dimensions to correctly align elements from each position. This property makes matrix addition a simple, entry-level operation compared to other matrix manipulations, like multiplication.

Matrix Commutativity

Commutativity refers to the ability to change the order of operations without altering the result. In mathematics, this property holds true for many operations, like addition; however, when it comes to matrix multiplication, it's usually not the case.

For matrices \(A\) and \(B\), the expression \( AB = BA \) is generally false. This lack of commutativity in matrix multiplication results from the structured way multiplication combines the rows of the first matrix with the columns of the second.

• Unless phenomena such as specific conditions or special matrices, like the identity matrix or scalar matrices, are involved, matrix multiplication is non-commutative.

When dealing with equations like \((A+B)^2 = A^2 + 2AB + B^2\), commutativity is necessary to cancel terms and simplify expressions. As ordinary matrices do not adhere to this property, we see discrepancies between the expected and actual outcomes.

Square Matrices

A square matrix is a matrix with the same number of rows and columns, such as a 2x2 matrix or a 3x3 matrix. This characteristic is significant in many matrix operations.

In the context of multiplication and other higher-level matrix operations, square matrices allow for processes like transposing, finding determinants, and inverses with well-established rules. For example, the identity matrix, which is central to matrix algebra, is a square matrix where all diagonal elements are 1 and all other elements are 0.

• Square matrices simplify notation and operations. For instance, the expressions \( A^2 \) and \( (A+B)^2 \) are only feasible if \( A \) and \( B \) are square and of the same size.
• These matrices also uniquely determine if a transformation is invertible (non-zero determinant), which is pivotal in linear algebra.

In exercises like comparing \((A+B)^2\) and the squared sum of matrices, working with square matrices is essential for ensuring that all dimension-related properties and operations are valid.

Matrix Properties

Matrices boast a wide collection of properties, vital for understanding their behavior under different operations like addition, multiplication, and transposition.

Some fundamental properties include:

• Distributive Property: This property applies to multiplication over addition, i.e., \(A(B+C) = AB + AC\).
• Associative Property: Essential for grouping operations, it applies to addition: \((A + B) + C = A + (B + C)\), and multiplication: \((AB)C = A(BC)\).
• Identity Element: For any square matrix \(A\), an identity matrix \(I\) exists such that \(AI = IA = A\). Identity matrices serve as the multiplicative equivalent of '1' in real numbers.
• Zero Matrix: This matrix contains only zeros and serves as the additive identity: \(A + 0 = A\).

Understanding these properties is crucial when performing matrix calculations or proving mathematical statements involving matrices. By mastering such properties, students can predict the outcomes of matrix operations, simplify expressions, and avoid common mishaps related to incorrect assumptions, like expecting commutative behavior from non-commutative processes.