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Join the ServerLia’s words silenced the clamor of the entire auditorium.
“A more complex journey?”
“A transfer?”
The mages looked at each other; the thoughts that had become clear due to the “price list” metaphor were once again shrouded in mist.
They could understand one price list, but how could two price lists be multiplied?
In their perception, this was something that made absolutely no sense.
Lia looked at the confused expressions of the people below her, especially the scrutiny in Master Albin’s eyes; she knew the most critical step had arrived.
“It seems that everyone no longer has objections to the concept of the ‘price list’.”
Lia’s voice rang out again, capturing everyone’s attention firmly.
“Now, let’s make a small change.”
With her magic, she erased the words “Price List” from the blackboard.
“We are no longer passengers; instead, we have transformed into the operators of these two carriage companies.”
This shift in identity piqued the interest of many.
“As operators, we care not only about how much fare we collect from passengers, but more importantly, how much cost we must incur to operate a segment of the route.”
Lia said, “Let us assume that the power for the carriages comes from two different magical materials, which we shall call ‘Magic Crystal No. 1’ and ‘Magic Crystal No. 2’.”
She wrote these two terms on the blackboard.
“Now, let us redefine the matrices.”
Lia pointed to the matrix on the left.
“This is no longer a price list, but a ‘Route Resource Consumption Table’.”
“The first row represents the routes departing from the ‘Tower of Truth.’ The second row represents the routes departing from the ‘Magic Association’.”
“The first column represents the quantity of ‘Magic Crystal No. 1’ consumed. The second column represents the quantity of ‘Magic Crystal No. 2’ consumed.”
Lia clearly marked the definitions of the rows and columns next to the matrix with chalk.
“Therefore, the number ‘1’ in the first row and first column means that for every run of a route departing from the ‘Tower of Truth,’ 1 unit of ‘Magic Crystal No. 1’ is consumed.”
“The number ‘7’ in the first row and second column represents that a route departing from the ‘Tower of Truth’ consumes 7 units of ‘Magic Crystal No. 2’.”
“Similarly, the second row (3, 8) represents that a route departing from the ‘Magic Association’ consumes 3 units of ‘Magic Crystal No. 1’ and 8 units of ‘Magic Crystal No. 2’.”
This explanation was slightly more complex than the previous “price list,” but everyone present was a mage; they had no trouble understanding the concept of resource consumption.
Everyone nodded in succession, jotting down the new definitions in their notebooks.
“Very good. Now let us look at Matrix II on the right.” Lia’s chalk moved to the other side.
“This is the ‘Resource Procurement Price Table’.”
“The first row represents the price of Magic Crystal No. 1. The second row represents the price of Magic Crystal No. 2.”
“The first column represents the procurement price at the Tower of Truth. The second column represents the procurement price at the Magic Association.”
She also marked the definitions next to the matrix.
“So, the number ‘2’ in the first row and first column represents that at the Tower of Truth, the price of Magic Crystal No. 1 is 2 copper coins per unit.”
“And the number ‘6’ in the second row and first column represents that at the Tower of Truth, the price of Magic Crystal No. 2 is 6 copper coins per unit.”
“The values (5, 4) in the second column represent the procurement prices of these two crystals at the Magic Association, respectively.”
At this point, a brand-new and clear physical meaning for the matrices was established.
One was a list of how many resources were needed to operate a route, and the other was how much those resources cost at different locations.
The entire auditorium was so quiet that a pin drop could be heard; everyone sensed that the moment to reveal the answer was approaching.
“Now, here is the question.” Lia’s voice rose a few decibels.
“As an operator, I need to know: if I choose to depart from the Tower of Truth and procure all resources locally there, what is the total cost to operate the route once?”
This question was very concrete, and everyone began performing mental calculations.
Lia did not provide the answer immediately but guided everyone through the thinking process.
“To calculate the total cost, we need to know two things: First, what resources do we need and in what quantities? Second, how much do these resources cost at the Tower of Truth?”
“Departing from the Tower of Truth corresponds to which row of our ‘Route Resource Consumption Table’ (Matrix I)?” she asked.
“The first row!” a voice answered immediately.
“Exactly right! Which means we need 1 unit of Crystal No. 1 and 7 units of Crystal No. 2,” Lia confirmed.
“Then, to procure these resources at the Tower of Truth, which column of the ‘Resource Procurement Price Table’ (Matrix II) should we reference?”
“The first column!” more people answered in unison.
“Exactly right! At the Tower of Truth, the price of Crystal No. 1 is 2, and Crystal No. 2 is 6.”
Lia wrote the calculation formula on the blackboard: Total Cost = (Quantity of Crystal No. 1 needed × Price of Crystal No. 1) + (Quantity of Crystal No. 2 needed × Price of Crystal No. 2).
She substituted the numbers as she wrote: Total Cost = (1 × 2) + (7 × 6).
“1 times 2 equals 2. 7 times 6 equals 42. 2 plus 42 equals 44.”
“Therefore, the total cost is 44 copper coins.”
Lia wrote down the final number, “44.”
The moment that number appeared, a collective gasp echoed throughout the grand auditorium.
44!
This was precisely the number in the top-left corner of the mysterious product matrix Lia had calculated earlier!
Heisenberg’s body jolted; he stared fixedly at the formula on the blackboard, his eyes erupting with a light of disbelief.
‘So that’s it! So that’s how it is!’
The “row times column” that had haunted him for countless days and nights—the physical logic behind it was actually so clear and obvious!
“Now, does everyone understand where this ‘row times column’ comes from?” Lia said with a smile.
“We multiply the corresponding elements of the first row of Matrix I and the first column of Matrix II, then add the results.”
“In our example, this process represents ‘calculating the total cost of procuring the resources required for a specific route at a specific location’.”
Master Albin’s mouth hung open; he felt his mind go blank.
He had originally prepared countless sharp questions, intending to challenge the rationality of this operational rule.
But now, the other party had used an impeccable business case to demonstrate the rationality of the rule to its fullest extent.
This was not witchcraft created out of thin air; this was logic! It was the logic of cost accounting that existed everywhere in the real world!
“Now, let us continue.” Lia’s voice pulled everyone back from their shock.
“If I still depart from the ‘Tower of Truth’ (the first row of Matrix I), but choose to procure all resources at the ‘Magic Association’ (the second column of Matrix II), what is the total cost?”
This time, without waiting for Lia’s guidance, many mages had already begun calculating for themselves.
“The quantity of Crystal No. 1 needed is 1, and its price at the Magic Association is 5; the quantity of Crystal No. 2 needed is 7, and its price at the Magic Association is 4.”
“Total Cost = (1 × 5) + (7 × 4) = 5 + 28 = 33!”
A young mage shouted the answer excitedly.
Lia nodded approvingly and wrote “33” on the blackboard.
“Exactly, 33. This is precisely the second number in our final result matrix.”
She rewrote the result matrix that she had erased earlier.
“Now, the first row is complete. It tells us that for the route departing from the ‘Tower of Truth,’ if resources are procured at the ‘Tower of Truth’ and the ‘Magic Association’ respectively, the total costs are 44 and 33.”
“So, the two question marks below—who can tell me what they are and what they represent?” Lia looked down at the audience.
The atmosphere in the hall was completely different this time.
It was no longer one of confusion and doubt, but of eager excitement.
“I know!” An apprentice in the front row stood up immediately. “The bottom is for calculating the costs for the route departing from the ‘Magic Association’! That corresponds to the second row of Matrix I!”
“If we procure from the ‘Tower of Truth,’ the cost is (3 × 2) + (8 × 6) = 54!”
“If we procure from the ‘Magic Association,’ the cost is (3 × 5) + (8 × 4) = 47!”
Following his words, Lia filled in the final two numbers on the blackboard.
This mathematical operation, which had once felt alien and grotesque to everyone, had now become incredibly clear and rational in their eyes.
It was no longer cold symbols, but a complete business model encompassing “Route Planning,” “Resource Consumption,” “Location,” “Price,” and “Total Cost.”
“Look,” Lia pointed to the final matrix, “this is matrix multiplication.”
“It combines two matrices containing different information—the consumption table and the price table—to yield a matrix containing brand-new information: the total cost table.”
“This is the ‘transfer.’ Through a single operation, we combine the steps of ‘what to do’ (the route) and ‘how to do it’ (the procurement) to directly derive the final result.”
“And so, it is not difficult to understand why AB ≠ BA.”
Lia turned around. “AB represents determining the route first and then considering procurement. But what about BA?”
She wrote the calculation for the first element: (2 × 1) + (5 × 3) = 17.
“What does this ’17’ represent? It represents (Price of Crystal No. 1 at ‘Tower of Truth’ × Consumption of Crystal No. 1 on the ‘Tower of Truth’ route) + (Price of Crystal No. 1 at ‘Magic Association’ × Consumption of Crystal No. 1 on the ‘Magic Association’ route).”
“In our business model, this is a meaningless jumble of numbers.”
“Different sequences represent entirely different operational processes; naturally, the results are vastly different. One has a clear physical meaning, while the other might be complete nonsense.”
Lia put down the chalk and looked around the room.
“Mister Heisenberg is telling us that the state of a microscopic particle is like our route. When we perform an observation on it, such as measuring its position, it is like completing a ‘transfer’; its state changes.”
“Measuring its momentum afterward is yet another ‘transfer’.”
“Measuring position then momentum versus measuring momentum then position are two completely different ‘journeys’; the final ‘destinations’ reached are naturally different.”
“This is the physical essence behind why matrix multiplication does not satisfy the commutative law.”
“It is not the tyranny of mathematics, but the grammar of the universe’s foundation. We have only just received our first ticket to a new world.”
As her voice fell, a long silence filled the grand auditorium.
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Read : I Possessed the Obsessive-Captive Dominant Top
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