Matrix algebra
Objective. Compose matrix operations with dimension checks.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Matrix algebra defines addition (entrywise, requiring equal dimensions), scalar multiplication, and matrix multiplication, where the product AB is defined only when the number of columns of A equals the number of rows of B, producing a matrix with A's row count and B's column count. Matrix multiplication is associative and distributive over addition but not generally commutative, so AB≠BA in general. For A a 2×3 matrix and B a 3×2 matrix, AB is 2×2 while BA is 3×3, entirely different sizes. Before composing any operation, checking dimension compatibility prevents undefined expressions.
Worked example
Given A=[[1,2],[3,4]] and B=[[0,1],[1,0]], compute the matrix product AB using matrix algebra.
- 1Row 1 of A times column 1 of B: (1)(0)+(2)(1)=2.
- 2Row 1 of A times column 2 of B: (1)(1)+(2)(0)=1.
- 3Row 2 of A times column 1 of B: (3)(0)+(4)(1)=4.
- 4Row 2 of A times column 2 of B: (3)(1)+(4)(0)=3, giving AB=[[2,1],[4,3]].
Answer: AB = [[2, 1], [4, 3]]
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students assume AB=BA as with real numbers, but matrix multiplication is generally non-commutative; even when both products are defined and the same size, they can give different matrices.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.