Multiply a Row by a Column
A matrix acts on an input column by taking each row’s dot product with that column.
About 25 minutes, plus any extended practice
Open lesson 1Precalculus: Matrices and Linear Systems
Matrices organize linear operations and systems through rows and columns. Multiplication uses row-column products, a determinant tests two-dimensional invertibility, and valid row operations preserve a system’s solution set when applied to every entry.
Before you begin
- Evaluate dot products; solve linear systems; multiply signed numbers.
Choose by skill rather than age alone. Read with an adult when useful, revisit a lesson, or explore another grade.
Follow the lesson sequence
Subtract the Cross Products in a Two-by-Two Determinant
For a two-by-two matrix, determinant is the upper-left times lower-right minus the upper-right times lower-left.
About 25 minutes, plus any extended practice
Open lesson 2Verify an Inverse by Producing the Identity
An inverse matrix undoes a matrix’s linear operation.
About 25 minutes, plus any extended practice
Open lesson 3Update the Entire Augmented Row
Subtracting a multiple of one equation from another corresponds to subtracting that multiple of an augmented row.
About 25 minutes, plus any extended practice
Open lesson 4Review and transfer
After the lessons, try 4 new review cases and a practical task. Each answer has an explanation and a link back to the related lesson.
Open the unit reviewGuest answers are temporary. Keep your own paper or document; this unit does not certify completion or record a school grade.
Topic references
- OpenStax mathematical topic reference: topic reference — Precalculus: Matrices and Linear Systems
These references informed the topic selection. They do not certify complete standards alignment.