State the dimensions of [[1,2,3],[4,5,6]].
Answer: 2×3
Reasoning: Two rows and three columns.
Reasoning: Two rows and three columns.
Teacher Key · Unit 11 · Lesson 01
Matrices, Dimensions & Operations
Use this key to verify method, notation, domain, units, and interpretation. Accept equivalent exact forms when mathematically valid.
Khaemenes Academy · Teacher Key
Matrices, Dimensions & Operations
Add [[1,2],[3,4]] and [[5,6],[7,8]].
Answer: [[6,8],[10,12]]
Reasoning: Add corresponding entries.
Reasoning: Add corresponding entries.
Can a 2×3 matrix multiply a 2×2 matrix on the right?
Answer: No
Reasoning: Inner dimensions 3 and 2 do not match.
Reasoning: Inner dimensions 3 and 2 do not match.
Compute [[1,0],[0,−1]]⟨3,4⟩.
Answer: ⟨3,−4⟩
Reasoning: This reflects across the x-axis.
Reasoning: This reflects across the x-axis.
What matrix represents a 90° counterclockwise rotation?
Answer: [[0,−1],[1,0]]
Reasoning: It maps ⟨x,y⟩ to ⟨−y,x⟩.
Reasoning: It maps ⟨x,y⟩ to ⟨−y,x⟩.
Why does order matter in matrix transformations?
Answer: Composition is generally noncommutative
Reasoning: AB and BA apply transformations in different orders.
Reasoning: AB and BA apply transformations in different orders.