Math Problem Statement

Prove that for matrices A = [3 2 1] and B = [-3 4 2], (A + B)^T = A^T + B^T.

Solution

To prove the property given in the problem, we need to verify that for the matrices A=[32]A = \begin{bmatrix} 3 & 2 \end{bmatrix} and B=[342]B = \begin{bmatrix} -3 & 4 & 2 \end{bmatrix}:

  1. (A+B)T=AT+BT(A + B)^T = A^T + B^T.

Step-by-step proof:

  1. Calculate the transpose of AA: AT=[32]A^T = \begin{bmatrix} 3 \\ 2 \end{bmatrix}

  2. Calculate the transpose of BB: BT=[342]B^T = \begin{bmatrix} -3 \\ 4 \\ 2 \end{bmatrix}

  3. Add matrices AA and BB: Since AA and BB are vectors/matrices of different shapes, A+BA + B is undefined, so the addition property cannot be applied directly.

This is a fundamental issue; matrix addition requires matrices of the same dimensions. Hence, the provided equation is invalid as the matrices are not conformable for addition.

If you have further questions or want to explore similar valid problems, please let me know.

Questions for further exploration:

  1. What conditions must be met for matrix addition and transposition properties to hold?
  2. How can the transposition operation be extended to higher-dimensional matrices?
  3. Can you explore examples of matrix properties such as distributivity with scalar multiplication?
  4. What are some real-world applications of matrix operations in computer graphics?
  5. How do we verify properties like (AB)T=BTAT(AB)^T = B^T A^T for conformable matrices?

Tip:

Always check the dimensions of matrices before performing operations like addition or multiplication to ensure they are conformable.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Matrix Transpose
Matrix Addition
Matrix Properties

Formulas

(A + B)^T = A^T + B^T

Theorems

Transpose of Matrix Addition

Suitable Grade Level

Grades 10-12