Question
Exercise 3 (15 points): Demonstrate the following properties, justifying: 1) If A, B and C are matrices of format m × n, then A + (B + C) = (A + B) + C. 2) If A is a matrix of format m × n and r, s are scalars, then (r + s)A = rA + sA.
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Answer to a math question Exercise 3 (15 points): Demonstrate the following properties, justifying: 1) If A, B and C are matrices of format m × n, then A + (B + C) = (A + B) + C. 2) If A is a matrix of format m × n and r, s are scalars, then (r + s)A = rA + sA.
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Solution:
1) To prove the property, we need to show that both sides of the equation, A + (B + C) and (A + B) + C, are equal.
Starting with the left-hand side, A + (B + C), we use the associativity of matrix addition which states that (A + B) + C = A + (B + C). Therefore, we can rewrite the left-hand side as:
A + (B + C) = (A + B) + C
This proves the property.
2) To prove the property, we need to show that both sides of the equation, (r + s)A and rA + sA, are equal.
Starting with the left-hand side, (r + s)A, we need to distribute the scalar (r + s) to the matrix A. Using the distributive property of scalar multiplication over matrix addition, we can rewrite the left-hand side as:
(r + s)A = rA + sA
This proves the property.
Answer:
1) The property is true: A + (B + C) = (A + B) + C.
2) The property is true: (r + s)A = rA + sA.
1) To prove the property, we need to show that both sides of the equation, A + (B + C) and (A + B) + C, are equal.
Starting with the left-hand side, A + (B + C), we use the associativity of matrix addition which states that (A + B) + C = A + (B + C). Therefore, we can rewrite the left-hand side as:
A + (B + C) = (A + B) + C
This proves the property.
2) To prove the property, we need to show that both sides of the equation, (r + s)A and rA + sA, are equal.
Starting with the left-hand side, (r + s)A, we need to distribute the scalar (r + s) to the matrix A. Using the distributive property of scalar multiplication over matrix addition, we can rewrite the left-hand side as:
(r + s)A = rA + sA
This proves the property.
Answer:
1) The property is true: A + (B + C) = (A + B) + C.
2) The property is true: (r + s)A = rA + sA.
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