Represent Systems as Matrix Equations

One Compact Notation for Every Linear System

In this lesson:

• Read and multiply matrices; access entries by index
• Write any system as ; convert back

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Read matrix dimensions; access entry
2. Compute 2×2 products as dot products
3. Write any system as ; name all three parts
4. Convert between system and matrix form, both ways
5. Explain the "same , different " advantage

One Notation for Every Linear System

Two equations, two unknowns. Could one compact object encode both?

What if we stored the coefficients in a table and wrote the whole system as a single equation?

Matrices: Notation and Entry Access

A matrix is a rectangular array of numbers. Dimensions: (rows × columns). Entry is in row , column .

Matrix Multiplication: Row Times Column

Each result entry is a dot product: row of with column of .

Not element-wise.

Computing a Full Matrix Product Step by Step

R1·C1:   R1·C2:

R2·C1:   R2·C2:

The Variable Vector Connects to Systems

Watch what happens when the second matrix is :

Those are exactly the left-hand sides of two linear equations.

Naming the Parts of

• : coefficient matrix — each row = one equation
• : variable vector; : constant vector

Converting a System to

System: and

Row 1 of : coefficients 3, 2. Row 2: 5, −4. Constants go in .

Verify: Reproduces the System

Row 1: ✓
Row 2: ✓

Reverse: Matrix Equation to System

Given:

Read each row of as one equation's coefficients:

Row 1:
Row 2:

Each row of → one equation.

Writing Zero Coefficients for Missing Variables

System: and

Rewrite equation 1 as :

Missing variable means coefficient is 0 — write it explicitly.

Common Error: You Cannot Divide by a Matrix

Wrong: — matrices can't be divided

Correct:

Scalar analogy: — multiply by the inverse.

Check-In: System to Matrix Form

Write each system as and identify , , :

1. and
2. and

For (2): rearrange to standard form first.

Same , Different — the Power

Same coefficient matrix encodes the structure of many systems:

Two different systems — one coefficient matrix. If exists, it solves both.

Factory Scenario: Structure vs. Specifics

A factory mixes two materials. Batch equations (fixed ratios ):

Week 1 targets . Week 2 targets .

Both weeks use the same . Only changes.

Your Turn: What Changes, What Stays?

. Write the matrix equation for:

What stays the same? What changes? What does each represent in the system?

Unscaffolded: Matrix Equation to System

Given:

Write the system of equations. Then compute to verify your system matches the matrix equation.

No hints. Show all steps.

The Identity Matrix — The Matrix "1"

The identity matrix leaves any matrix or vector unchanged — like multiplying by 1.

If exists: .

Scalar Analogy: Solving

Multiply on the LEFT — order matters.

What You Can Do Now

✓ Access ; compute 2×2 matrix products

✓ Write any system as ; name all three parts

✓ Convert matrix equation → system; verify

ROW of = one equation — not columns

Zero coefficients must appear explicitly

Multiply by on the LEFT

Coming Up: Finding and Using

Next: HSA.REI.C.9 — find for matrices; solve completely.

Key question: when does exist? Not always — connected to the determinant.

REI.C.8 gave us the form. REI.C.9 gives us the solution method.