Math Problem Statement
π΄ , π΅, and πΆ are three matrices of compatible sizes so that the products π΄π΅ and π΄π΅πΆ 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 π statements which are true in a list of π questions. Suppose you select π1 statements which are true, and you select π2 statements which are false. Then, the number of points you get is ππ1πβπ2 out of a max total of π possible points. Question 1 Answer
a. If rows 1 and 3 of π΅ are the same, so are rows 1 and 3 of π΄π΅.
b. If columns 1 and 3 of π΅ are the same, so are the columns 1 and 3 of π΄π΅.
c. If rows 1 and 3 of π΄ are the same, so are rows 1 and 3 of π΄π΅πΆ.
d. If π΄π΅ and π΅π΄ are both defined, then π΄π΅ and π΅π΄ are both square (though possibly of different sizes).
e. (π΄π΅)2=π΅2π΄2.
Solution
Let's analyze each statement one by one, providing reasoning or counterexamples as needed:
a. If rows 1 and 3 of π΅ are the same, so are rows 1 and 3 of π΄π΅.
- True. When multiplying two matrices, the rows of the resulting matrix are linear combinations of the rows of , with the coefficients being the elements of . If rows 1 and 3 of are the same, then the linear combinations (i.e., the rows) in the product matrix will also be the same because the coefficients for rows 1 and 3 in are identical. Therefore, rows 1 and 3 of will also be the same.
b. If columns 1 and 3 of π΅ are the same, so are the columns 1 and 3 of π΄π΅.
- False. While the columns of the resulting product matrix are linear combinations of the columns of (weighted by the corresponding rows of ), the fact that columns 1 and 3 of are the same doesn't necessarily imply that columns 1 and 3 of will be the same. This is because the rows of can affect the way the columns of are combined. A counterexample: if has non-identical rows, the linear combinations might not preserve the equality between columns 1 and 3 of .
c. If rows 1 and 3 of π΄ are the same, so are rows 1 and 3 of π΄π΅πΆ.
- True. In the product , the rows of are used as the basis for the rows of the resulting product. If rows 1 and 3 of 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 will also be the same.
d. If π΄π΅ and π΅π΄ are both defined, then π΄π΅ and π΅π΄ are both square (though possibly of different sizes).
- False. If and are both defined, the number of rows and columns of and must align appropriately. However, this does not imply that the resulting matrices must be square. The product could result in a rectangular matrix, and the product could be of a different rectangular shape. A counterexample: Let be a matrix and be a matrix. Both (which results in a matrix) and (which results in a matrix) are defined, but neither is square.
e.
- False. In general, matrix multiplication is not commutative, meaning that in most cases. Therefore, squaring will not give the same result as squaring . In fact, , which is not the same as 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:
- What happens when both rows and columns of two matrices are identical?
- How does matrix multiplication affect the symmetry of matrices?
- Can two non-square matrices yield a square product?
- Under what conditions can the matrix multiplication hold?
- 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
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