Math Problem Statement

𝐴 , 𝐡, and 𝐢 are three matrices of compatible sizes so that the products 𝐴𝐡 and 𝐴𝐡𝐢 are defined. In the following statements, when we say that a matrix has rows (resp. columns) 1 and 3 the same, we assume that this matrix has enough rows (resp. columns) in order for there to be a first and third row (resp. column). Select all the statements below that are true. Hint: For each statement, try testing some examples. In an exam, you will be asked to give a reason if the statement is True and a counterexample if the statement is False. Grading: Here is how these types of questions are graded. Say there are 𝑛 statements which are true in a list of 𝑁 questions. Suppose you select π‘š1 statements which are true, and you select π‘š2 statements which are false. Then, the number of points you get is π‘π‘š1π‘›βˆ’π‘š2 out of a max total of 𝑁 possible points. Question 1 Answer

a. If rows 1 and 3 of 𝐡 are the same, so are rows 1 and 3 of 𝐴𝐡.

b. If columns 1 and 3 of 𝐡 are the same, so are the columns 1 and 3 of 𝐴𝐡.

c. If rows 1 and 3 of 𝐴 are the same, so are rows 1 and 3 of 𝐴𝐡𝐢.

d. If 𝐴𝐡 and 𝐡𝐴 are both defined, then 𝐴𝐡 and 𝐡𝐴 are both square (though possibly of different sizes).

e. (𝐴𝐡)2=𝐡2𝐴2.

Solution

Let's analyze each statement one by one, providing reasoning or counterexamples as needed:


a. If rows 1 and 3 of 𝐡 are the same, so are rows 1 and 3 of 𝐴𝐡.

  • True. When multiplying two matrices, the rows of the resulting matrix AΓ—BA \times B are linear combinations of the rows of AA, with the coefficients being the elements of BB. If rows 1 and 3 of BB are the same, then the linear combinations (i.e., the rows) in the product matrix AΓ—BA \times B will also be the same because the coefficients for rows 1 and 3 in BB are identical. Therefore, rows 1 and 3 of AΓ—BA \times B will also be the same.

b. If columns 1 and 3 of 𝐡 are the same, so are the columns 1 and 3 of 𝐴𝐡.

  • False. While the columns of the resulting product matrix AΓ—BA \times B are linear combinations of the columns of BB (weighted by the corresponding rows of AA), the fact that columns 1 and 3 of BB are the same doesn't necessarily imply that columns 1 and 3 of AΓ—BA \times B will be the same. This is because the rows of AA can affect the way the columns of BB are combined. A counterexample: if AA has non-identical rows, the linear combinations might not preserve the equality between columns 1 and 3 of AΓ—BA \times B.

c. If rows 1 and 3 of 𝐴 are the same, so are rows 1 and 3 of 𝐴𝐡𝐢.

  • True. In the product AΓ—BΓ—CA \times B \times C, the rows of AA are used as the basis for the rows of the resulting product. If rows 1 and 3 of AA are the same, then for any matrix multiplication afterward (with compatible matrices), these rows will remain the same in the final product. Thus, rows 1 and 3 of AΓ—BΓ—CA \times B \times C will also be the same.

d. If 𝐴𝐡 and 𝐡𝐴 are both defined, then 𝐴𝐡 and 𝐡𝐴 are both square (though possibly of different sizes).

  • False. If AΓ—BA \times B and BΓ—AB \times A are both defined, the number of rows and columns of AA and BB must align appropriately. However, this does not imply that the resulting matrices must be square. The product AΓ—BA \times B could result in a rectangular matrix, and the product BΓ—AB \times A could be of a different rectangular shape. A counterexample: Let AA be a 2Γ—32 \times 3 matrix and BB be a 3Γ—23 \times 2 matrix. Both AΓ—BA \times B (which results in a 2Γ—22 \times 2 matrix) and BΓ—AB \times A (which results in a 3Γ—33 \times 3 matrix) are defined, but neither is square.

e. (AΓ—B)2=B2Γ—A2.(A \times B)^2 = B^2 \times A^2.

  • False. In general, matrix multiplication is not commutative, meaning that AΓ—Bβ‰ BΓ—AA \times B \neq B \times A in most cases. Therefore, squaring AΓ—BA \times B will not give the same result as squaring BΓ—AB \times A. In fact, (AΓ—B)2=AΓ—BΓ—AΓ—B(A \times B)^2 = A \times B \times A \times B, which is not the same as B2Γ—A2B^2 \times A^2 unless special conditions hold (like commutative matrices, which is rare in practice).

Summary of answers:

  • True statements: a, c
  • False statements: b, d, e

Would you like more detailed explanations or have any follow-up questions?

Here are five related questions for further exploration:

  1. What happens when both rows and columns of two matrices are identical?
  2. How does matrix multiplication affect the symmetry of matrices?
  3. Can two non-square matrices yield a square product?
  4. Under what conditions can the matrix multiplication (AΓ—B)2=B2Γ—A2(A \times B)^2 = B^2 \times A^2 hold?
  5. How does the rank of a matrix change after multiplying with another matrix?

Tip: Always verify matrix dimensions before performing matrix multiplication to ensure compatibility and avoid errors in calculations!

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Math Problem Analysis

Mathematical Concepts

Matrix Multiplication
Linear Algebra
Matrix Properties

Formulas

Matrix Product AB
Matrix Product ABC

Theorems

Linear Combination of Matrix Rows
Matrix Multiplication Commutativity (or lack thereof)

Suitable Grade Level

Grades 11-12