Use Matrices to Represent Data

HSN.VM.C.6 — Lesson 6 of 12

1. State dimensions and use vocabulary: row, column, entry
2. Locate using row-column index notation
3. Encode tabular data as a matrix
4. Construct and read a payoff matrix
5. Construct and interpret an incidence matrix

What Does Position Tell Us?

What question could you NOT answer from this grid alone?

Hint: the numbers are there — what's missing?

The Grid Gets a Context

What Is a Matrix? Definition and Notation

A matrix is a rectangular array of numbers in brackets.

• Rows go across; columns go down
• Dimensions: (rows × columns)
• is : 2 rows, 3 columns

Dimensions: Rows × Columns (RC Rule)

• 3 horizontal strips → 3 rows
• 4 vertical strips → 4 columns
• Mnemonic: RC — row, then column

Locating Any Entry Using Row and Column

= row , col — row first, always.

• : row 1, col 2
• : row 2, col 3
• : row 1, col 1

Three Special Matrix Shapes Worth Knowing

• Square: equal rows and columns
• Column vector: same as VM.A.2 component form

Worked Drill: Locate Four Specific Entries

| | |

For each: state the row, then scan to the column.

Check-In: Three Skills at Once

1. What are the dimensions of ?
2. Find
3. Find

Assigning Meaning to Rows and Columns

Rows = Products; Columns = Weeks

: Product B, Week 3. C's total → row 3. Week 2 → col 2.

Encoding Data as a Matrix

The table and the matrix are the same data — different notation.

Payoff Matrix: Two Players, Two Choices

P1 picks a row; P2 picks a column. Entry = P1's payoff.

• : same strategy → P1 wins
• : different strategies → P1 loses
• : different strategies → P1 loses

Reading Payoff Entries in Context

• P1=Down, P2=Left →
• Worst-case per row: both mins are
• No row dominates — neither strategy is always better

From Quantities to Binary: 0 or 1

In a directed network, each connection is either:

• Present → encode as 1
• Absent → encode as 0

No quantities, just connectivity. The same matrix structure holds.

"Can you fly directly from City A to City C?" is a matrix lookup.

Incidence Matrix: Networks in a Grid

if there is a directed edge from node to node ; else .

Building an Incidence Matrix: Worked

A→B, A→C, B→D, C→B, D→A

Find and Fix the Dimension Error

A student wrote:

" has dimensions ."

Identify the error. State the correct dimensions and explain.

Full Practice: Matrix to Network and Back

Network: C1→C2, C1→C3, C2→C4, C3→C2, C4→C1, C4→C3

1. Write the incidence matrix
2. From C3, which cities are directly reachable?
3. Is symmetric? What would that imply about the roads?

Edge Case: Zero Means "No Connection"

means no edge — not missing data.

• Zero is a valid, meaningful entry
• "No units sold" and "no connection" are facts, not gaps

Every position holds a value. Zero is information, not absence.

Edge Case: Directed Means Asymmetric

One-way: exists, does not.

• (edge exists)
• (no reverse edge)

Directed networks are not symmetric — row ≠ row .

Key Takeaways from This Lesson

✓ RC: rows × columns — rows always first

✓ Entry: = row , col

✓ Data matrix: position carries category meaning

✓ Payoff: row = P1; col = P2; entry = P1's payoff

✓ Incidence: if edge exists; else

Next Lesson: Scaling an Entire Matrix

VM.C.7: Multiply every entry by scalar :

Same entry-by-entry rule as VM.B.5 — applied to the whole matrix.