Week 11 · Matrices

Friday

Matrices everywhere

// The lane card grid

⏱ about 20 min

Friday: Matrices Everywhere

Friday the crew clears the corkboard and lists every grid they have met this week.

"The lane tallies," Comet says. "Runs by boat and course. Laps and turns per run."

"The hull's corners," Wren adds. "And the stretch, the flip, the shear. Every one a matrix."

Nova projects a map of the three docks with arrows between them. "Would you like a hint? A grid can hold a network."

"Row for the dock you leave, column for the dock you reach," Wren says. "A 1 if a path exists, a 0 if not."

"So a matrix holds counts, moves and connections," Comet says. "What do you notice they share?"

"Rows and columns with meanings," Wren says. "Keep the labels and the arithmetic stays honest."

"Then one more time through everything," Comet says, "and the log is complete."

Where matrices show up

Tables of counts: tallies by heat and lane, runs by boat and course. Add days, scale plans.
Rates per item: laps and turns per run. Multiply counts by rates to get totals, labels matching in the middle.
Transformations: a 2 by 2 matrix stretches, flips, shears or turns every point of a drawing.
Networks: a 1 where two docks are joined by a path and a 0 where they are not. The grid is the map.
Systems of equations: two unknowns, two equations, one matrix sentence solved with an inverse.

A dock network

Three docks: the shed dock, the far post and the buoy raft. A path runs shed to post, post to raft and shed to raft.

Rows are where you leave, columns are where you arrive. Each path goes both ways, so the matrix is symmetric.

$The dock network as a 3 by 3 grid: a 1 where a path joins two docks and 0 on the diagonal.$

Each row adds to 2: every dock is joined to two others. The matrix holds the whole map, and no picture is needed to read it.

Mixed review

FROM THE LOG

Read the question.
Tap your answer.

Two days of tallies: [1 3; 2 0] and [4 1; 1 5]. What is their sum?

Multiply [2 1; 0 3] × [1 2; 4 0].

$A blue shape with vertices (0, 0), (3, 2) and its red image under [1 0; 0 -1], vertices (0, 0), (3, -2)$ Apply the flip [1 0; 0 -1] to the point ⟨3, 2⟩. Where does it land?

$A blue shape with vertices (0, 0), (1, 0), (1, 1), (0, 1) and its red image under [3 0; 0 2], vertices (0, 0), (3, 0), (3, 2), (0, 2)$ The matrix [3 0; 0 2] transforms the hull drawing. By what factor does it multiply every area?

REASON IT OUT

Read the question.
Tap your answer.

A 2 by 3 matrix is multiplied by a 3 by 2 matrix. What size is the product?

A matrix equation A[x; y] = b has a unique solution. What must be true of A?

Which matrix leaves every point of the plane where it is?

A transformation matrix has determinant -4. What happened to the hull?

In the dock network matrix, how many 1s are in the Shed row? Type the number.

WHY THIS EXERCISEA row sum in a network matrix counts connections. Matrices hold maps as easily as tallies.

Statement	True or false?
Matrices of the same size add entry by entry.	?
A scalar multiplies only the diagonal entries.	?
2 × 1 + 1 × 4 = 6	?
The determinant's absolute value is the area factor of the transformation.	?
A matrix with determinant 0 has an inverse.	?

WHY THIS EXERCISEThe review mixes every move from the week: add, scale, multiply, transform, determinant and inverse.

Try it

On paper, draw three or four places in your home as dots with paths between them. Write the network matrix.
Add each row. Which place is joined to the most others?

Boathouse Log: list five places a matrix showed up this week and the size of each.Write the three rules for multiplying matrices: sizes, row times column, and order.Write one matrix from your own home, a tally or a network, with its labels.Write one sentence on what the determinant told you this week, and one on what the inverse did.

Fine week's work. Tomorrow is Dock Day: the boat cutout goes on the family table.

Grown-up checked my answers

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