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Precalculus 9-12 / Week 11 / Monday
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Week 11 · Matrices

Monday

A table with a name
// The lane card grid
⏱ about 20 min

Monday: A Table With a Name

The corkboard in the shed is a grid of lane cards. Heats down the side, lanes across the top.

"Every card has a tally," Wren says. "Morning practice runs. Heat A, lane 1, three runs."

"Six cards, six numbers," Comet says. "What can we make of a grid like that?"

"Give it a name," Wren says, writing M beside the board. "Then the whole grid is one object."

Nova projects the six numbers in square brackets. "Would you like a hint? Rows first, then columns."

"Two rows, three columns," Comet reads. "A 2 by 3 matrix."

"The afternoon tally is another one, the same size," Wren says. "Add them card by card and we have the day."

"And double every card for the two-week plan," Comet says. "That is one number times the whole grid."

The corkboard grid of lane cards in the shed, with Wren sliding a boat cutout to a stretched, flipped position.

What a matrix is

A matrix is a rectangular table of numbers. Each number is an entry, and each entry has a row and a column.

The morning tally is M = [3 2 1; 1 4 2]. Rows are separated by semicolons: the first row is 3, 2, 1.

M has 2 rows and 3 columns, so it is a 2 by 3 matrix. Rows first, then columns, always.

The entry in row 2, column 2 is 4: Heat B ran lane 2 4 times.

The morning tally as a 2 by 3 grid of lane cards, two heats down and three lanes across.

Adding matrices and scaling them

Two matrices of the same size add entry by entry: row 1 column 1 with row 1 column 1, and so on.

Afternoon tally N = [2 1 3; 0 2 1]. The day's total is M + N = [5 3 4; 1 6 3].

A scalar multiplies every entry. Doubling the morning plan for two weeks gives 2M.

2M = [6 4 2; 2 8 4]. Every card doubled, nothing else changed.

Matrices of different sizes cannot be added. A 2 by 3 and a 3 by 2 have no matching entries.

A solved problem to study

  1. The crew wants the day's total for Heat A, lane 3. M has 1 there and N has 3.
  2. Add the matching entries: 1 + 3 = 4.
  3. Do the same for all six positions: M + N = [5 3 4; 1 6 3].
  4. Now the two-week plan: multiply each entry of M by 2. Row 1 becomes 6, 4, 2.
  5. So 2M = [6 4 2; 2 8 4]. The size stays 2 by 3, because scaling moves no entries.
  6. Check one entry back: row 2, column 1 of 2M should be twice row 2, column 1 of M. It is.
MatrixMeaningRow 1Row 2
Mmorning runs3 2 11 4 2
Nafternoon runs2 1 30 2 1
M + Nthe day's runs5 3 41 6 3
2Mtwo weeks of mornings6 4 22 8 4
ADD AND SCALE THE TALLIES
  • Read the question.
  • Tap your answer.
Morning M = [3 2 1; 1 4 2] and afternoon N = [2 1 3; 0 2 1]. What is M + N? (Rows are separated by semicolons.)
The two-week plan doubles every morning count. What is 2 × [3 2 1; 1 4 2]?
A smaller board holds [1 2; 0 4]. Triple every entry. What is 3 × [1 2; 0 4]?
In M = [3 2 1; 1 4 2], what is the entry in row 2, column 2? Type the number.
WHY THIS EXERCISEEvery matrix skill this week starts with finding one entry by its row and its column.
StatementTrue or false?
M = [3 2 1; 1 4 2] is a 2 by 3 matrix.?
A 2 by 3 matrix can be added to a 3 by 2 matrix.?
1 + 3 = 4?
Multiplying a matrix by 2 doubles only the first row.?
2 × 4 = 8?
WHY THIS EXERCISEAdding and scaling are the easy moves. Tomorrow's multiplication is the one with a twist.
Try it
Make a 2 by 3 tally of your own: two rows for two days, three columns for three tasks.
Write a second tally the same size and add them. Then double the first one.
Draw the corkboard as a 2 by 3 grid of cards. Write the morning counts on them and label the rows and columns.

Strong start. Tomorrow a row meets a column, and matrix multiplication turns runs into laps.