Determinant Designs

Bridging Geometry and Linear Algebra

Pre-Calculus

Warm-up: The 2×2

Calculate the determinant of matrix A:

\( A = \begin{bmatrix} 4 & -2 \\ 3 & 5 \end{bmatrix} \)

Quick Recall

"The product of the main diagonal minus the product of the off diagonal."

det(A) = ad - bc

Warm-up: The 3×3 Challenge

Find the determinant using Sarrus' Rule:

\( M = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \)

Rewrite the first two columns to the right.

Sum the products of the three "downward" diagonals.

Subtract the products of the three "upward" diagonals.

Area of a Triangle

Coordinate Geometry Demonstration

WATCH CLOSELY

Focus 1:33

Note the specific expanded formula presented.

Focus 3:09

Watch the sign handling (double negatives!).

Focus 3:45

Why do we use the absolute value at the end?

The "Shoelace" Expansion

From the video, for vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \):

AREA FORMULA

Area = \( \frac{1}{2} |x_1y_2 + x_2y_3 + x_3y_1 - x_1y_3 - x_2y_1 - x_3y_2| \)

The first 3 terms are products of the Main Diagonals.

The last 3 terms are products of the Off Diagonals.

The Matrix Link

Standard Pre-Calculus Area Formula:

Area = \( \frac{1}{2} \left| \det \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \right| \)

Why is the third column all ones?

Lab Challenge

Does the Expansion by Minors of this matrix match the formula from the video?

Task

• Pick Row 1 to expand across.
• Distribute the \(x_1\) and \(y_1\) values.
• Group the positive and negative terms.

Use the Worksheet Part 2 to complete your derivation!

Polygons & Patterns

What if the polygon had 4 sides?

\( (x_1, y_1) \)

\( (x_2, y_2) \)

\( (x_3, y_3) \)

\( (x_4, y_4) \)

\( (x_1, y_1) \)

Prediction

How would the expansion formula change?

Would we need a 4x4 matrix with a column of 1s? Or is there something simpler about the "Shoelace" layout?

Exit Strategy

"Explain why the absolute value in the Shoelace formula is equivalent to the absolute value of a matrix determinant."

Write it down

Discuss with peer

Turn in worksheet

Determinant Designs

Determinant Designs Facilitation Guide

Determinant Designs Slides

Shoelace Strategy Anchor Chart

Coordinate Combat Worksheet

Coordinate Combat Answer Key

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