Matrix operations and systems
Check dimensions, perform matrix operations and interpret a product or matrix equation.
What you will tackle
- Add/subtract matrices and use scalar multiplication
- Multiply matrices using row–column products and dimension rules
- Recognise order dependence and identity
- Represent a linear system with a coefficient matrix
Match the operation to its rule
Addition and subtraction require equal dimensions and combine corresponding entries. Scalar multiplication scales each entry. An m×n matrix can multiply an n×p matrix to produce an m×p result. Each entry is a row–column dot product. Matrix multiplication generally is not commutative.
Matrices can encode several constraints
The system 2x + y = 7 and x − y = 2 is represented by [[2,1],[1,−1]][x,y]ᵀ = [7,2]ᵀ. The entries preserve coefficient position. The identity matrix leaves a compatible vector unchanged; a matrix equation does not justify dividing corresponding entries as though they were unrelated scalar equations.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
For A = [[1,2],[3,4]] and B = [[2,0],[−1,5]], find A + B, 2A and the first row of AB.
Show a starting hint
A row is multiplied against a column, not against the same-position entry.
Compare with the explained answer
- Add entries to obtain [[3,2],[2,9]].
- Scale to get [[2,4],[6,8]].
- The first row product is [1·2 + 2(−1), 1·0 + 2·5] = [0,10].
Answer / response criteria: A + B = [[3,2],[2,9]]; 2A = [[2,4],[6,8]]; first row of AB is [0,10].
Guided 2
For A = [[1,2],[3,4]] and B = [[2,0],[−1,5]], find A − B and explain why equal dimensions are required.
Compare with the explained answer
- Subtract corresponding entries: 1−2,2−0,3−(−1),4−5.
- The result is [[−1,2],[4,−1]].
- Equal dimensions provide a matching entry for every subtraction.
Answer / response criteria: [[−1,2],[4,−1]].
Independent practice
Independent 1
A is 2×3 and B is 3×1. State the dimensions of AB and whether BA is defined. Solve the system represented by [[2,1],[1,−1]][x,y]ᵀ = [7,2]ᵀ.
Compare with the explained answer
- The shared inner dimension is 3, so AB is 2×1.
- BA would multiply 3×1 by 2×3; inner dimensions 1 and 2 differ, so it is undefined.
- Adding 2x + y = 7 and x − y = 2 gives x = 3, then y = 1.
Answer / response criteria: AB is 2×1; BA is undefined; (x,y) = (3,1).
Independent 2
Find 2A − B for A = [[2,−1],[0,3]] and B = [[1,4],[−2,5]].
Compare with the explained answer
- 2A = [[4,−2],[0,6]].
- Subtract B entry by entry.
- The result is [[3,−6],[2,1]].
Answer / response criteria: [[3,−6],[2,1]].
Application practice
Application 1
A shop's daily sales counts are rows [3,2] and [1,5] for two products. Prices form column [4,6]. Find each day's revenue and explain the dimensions and units.
Compare with the explained answer
- Multiply the 2×2 count matrix by the 2×1 price column.
- Day 1 revenue is 3·4 + 2·6 = 24; day 2 is 1·4 + 5·6 = 34.
- The 2×1 result lists daily currency amounts because counts multiply currency per item.
Answer / response criteria: Revenue column [24,34]ᵀ; dimensions 2×1.
Application 2
A two-day, two-product sales matrix is [[8,5],[6,9]]. Returns are [[1,0],[2,1]]. Find net units and, at prices [3,4]ᵀ, net daily revenue.
Compare with the explained answer
- Subtract returns to get [[7,5],[4,8]].
- First-day revenue is 7·3 + 5·4 = 41.
- Second-day revenue is 4·3 + 8·4 = 44.
Answer / response criteria: Net matrix [[7,5],[4,8]]; revenues [41,44]ᵀ.
What can you explain now?
A two-day, two-product sales matrix is [[8,5],[6,9]]. Returns are [[1,0],[2,1]]. Find net units and, at prices [3,4]ᵀ, net daily revenue.
Compare with the explanation
Net matrix [[7,5],[4,8]]; revenues [41,44]ᵀ.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide