Section 22.2

Review: Matrices & Vectors

To solve systems efficiently, we pack numbers into grids.

1

Introduction

A Matrix is a rectangular array of numbers. A system of equations uses a matrix and vectors .

2

Matrix Multiplication

Row by Column

To find entry in the product , dot the -th row of A with the -th column of B.

3

Determinants

The determinant tells us if a unique solution exists. If , the matrix is "singular" and not invertible.

  • 2x2: .
  • 3x3: Cofactor Expansion (cross out row/col and multiply by sub-determinant).
4

Matrix Inversion

is the matrix such that .

For 2x2: .

5

Worked Examples

Example 1: Multiplication

Multiply by .

1. Procedure:

Row 1 dot Column 1: .

Row 2 dot Column 1: .

2. Result:

.

Example 2: Determinant

Find for .

1. Formula:

.

.

.

Example 3: Derivatives of Matrices

If , find .

Differentiation is element-wise.

.

6

Practice Quiz

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