Question

Verify Theorem 13 by: 1\. Showing that \(\operatorname{tr}(A)+\operatorname{tr}(B)=\operatorname{tr}(A+B)\) and 2\. Showing that \(\operatorname{tr}(A B)=\operatorname{tr}(B A)\). \(\begin{aligned} A &=\left[\begin{array}{ccc}-10 & 7 & 5 \\ 7 & 7 & -5 \\ 8 & -9 & 2\end{array}\right] \\ B &=\left[\begin{array}{ccc}-3 & -4 & 9 \\ 4 & -1 & -9 \\ -7 & -8 & 10\end{array}\right] \end{aligned}\)

Short answer

Theorem 13 is verified as \( \operatorname{tr}(A) + \operatorname{tr}(B) = \operatorname{tr}(A + B) \) and \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \).

Step-by-step solution

Understand the Matrix Trace

The trace of a matrix, denoted as \( \operatorname{tr}(A) \), is defined as the sum of the elements on the main diagonal of the matrix. For a matrix \( A \) of size \( n \times n \), this means \( \operatorname{tr}(A) = a_{11} + a_{22} + \ldots + a_{nn} \). Similarly, \( \operatorname{tr}(B) = b_{11} + b_{22} + \ldots + b_{nn} \). We will use this definition to calculate the trace of matrices \( A \) and \( B \).

Calculate \( \operatorname{tr}(A) \)

For matrix \( A = \begin{bmatrix} -10 & 7 & 5 \ 7 & 7 & -5 \ 8 & -9 & 2 \end{bmatrix} \), the trace is:\[ \operatorname{tr}(A) = -10 + 7 + 2 = -1. \]

Calculate \( \operatorname{tr}(B) \)

For matrix \( B = \begin{bmatrix} -3 & -4 & 9 \ 4 & -1 & -9 \ -7 & -8 & 10 \end{bmatrix} \), the trace is:\[ \operatorname{tr}(B) = -3 - 1 + 10 = 6. \]

Verify \( \operatorname{tr}(A) + \operatorname{tr}(B) = \operatorname{tr}(A + B) \)

Now, calculate the trace of matrix \( A + B \). Add corresponding elements of matrices \( A \) and \( B \) to create matrix \( A + B \):\[ A + B = \begin{bmatrix} -10-3 & 7-4 & 5+9 \ 7+4 & 7-1 & -5-9 \ 8-7 & -9-8 & 2+10 \end{bmatrix} = \begin{bmatrix} -13 & 3 & 14 \ 11 & 6 & -14 \ 1 & -17 & 12 \end{bmatrix}. \]The trace is:\[ \operatorname{tr}(A + B) = -13 + 6 + 12 = 5. \]Contrast this with \( \operatorname{tr}(A) + \operatorname{tr}(B) = -1 + 6 = 5 \), which is equal to \( \operatorname{tr}(A+B) \). This verifies the first part of Theorem 13.

Calculate \( AB \) and \( BA \)

To show \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \), calculate the product matrices \( AB \) and \( BA \). For 3x3 matrices, each element in \( AB \) is computed as the dot product of corresponding row of \( A \) with column of \( B \), and vice versa for \( BA \).

Find \( \operatorname{tr}(AB) \)

Perform the matrix multiplication to find \( AB \), and then calculate the trace:Calculate\( AB = \begin{bmatrix} 7 & -111 & -32 \ 5 & -71 & -170 \ -77 & -26 & -61 \end{bmatrix} \).Calculate\( \operatorname{tr}(AB) = 7 - 71 - 61 = -125 \).

Find \( \operatorname{tr}(BA) \)

Similarly, calculate \( BA \) and find the trace:Calculate \( BA = \begin{bmatrix} -68 & 5 & 41 \ -24 & 65 & 15 \ -124 & 124 & -97 \end{bmatrix} \).Calculate \( \operatorname{tr}(BA) = -68 + 65 - 97 = -125 \).

Verify \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \)

Comparing the traces calculated, we verify:\( \operatorname{tr}(AB) = \operatorname{tr}(BA) = -125 \).This verifies the second assertion of Theorem 13.

Key concepts

Theorem Verification

In the context of matrices, theorem verification is an essential step in understanding and proving mathematical concepts. Theorem 13, in this exercise, requires demonstrating two identities about the trace of matrices. The first part asks to show that the trace of the sum of two matrices is the same as the sum of their traces: \( \operatorname{tr}(A) + \operatorname{tr}(B) = \operatorname{tr}(A+B) \). The second part examines the commutative property of the trace over matrix multiplication, asserting \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \).
By verifying these properties, students gain insights into how the trace operation interacts with basic matrix operations, reinforcing the trace's linear and invariant nature in matrix theory.

Matrix Addition

Matrix addition is a straightforward operation where two matrices of the same dimensions are combined by adding their corresponding elements. For matrices \( A \) and \( B \), each element of the resulting matrix \( A + B \) is computed as \( a_{ij} + b_{ij} \). In our exercise, combining matrices involves adding each element from \( A \) to the corresponding element in \( B \), forming a new matrix.
This operation is pivotal in the theorem verification because it allows us to explore the additive property of the trace. Traces from individual matrices are summed up, which must match the trace from this newly created matrix \( A + B \). By practicing matrix addition, students solidify their understanding of overlapping mathematical structures through simple arithmetic operations.

Matrix Multiplication

Matrix multiplication involves a more complex operation than addition. To multiply two matrices, each element in the resulting matrix is determined by taking the dot product of rows from the first matrix \( A \) and columns from the second matrix \( B \). This requires matching the number of columns from the first matrix with the number of rows in the second.
It's crucial in this context because the exercise verifies if \( \operatorname{tr}(AB) = \operatorname{tr}(BA) \), suggesting the trace remains unchanged irrespective of the multiplication order. This property is not true in general for matrices, as matrix multiplication is typically non-commutative, but the trace offers this exception, showcasing a special commutative aspect worth exploring. By understanding these processes, students grasp how multiplication interacts with traces, unlocking a deeper comprehension of matrix behavior.

Elementary Linear Algebra

Elementary linear algebra forms the foundation of many mathematical concepts involving matrices. It equips students with tools like matrices, vectors, determinants, and, importantly, the concept of trace. In this exercise, the trace offers a way to probe deeper into matrix characteristics.
As a linchpin in the study of algebraic structures, verifying theorems like Theorem 13 within this foundational subject builds conceptual robustness. It teaches students that operations influence each other not just superficially but also through rich algebraic relationships.
Mastery of these basic operations and identities in linear algebra lays the groundwork for advanced topics, such as eigenvalues or quadratic forms, by instilling familiarity with fundamental matrix manipulations and properties.