MathematicsChapter 2: Matrices and Determinants
Addition and Scalar Multiply
Add (or subtract) entrywise, only if the sizes match.
Learning objectives
- State the main ideas and notation used in addition and scalar multiply.
- Apply the rules of addition and scalar multiply in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Add (or subtract) entrywise, only if the sizes match. Scalar k times A multiplies every entry by k. Negative: -A = (-1)A. Meaning: addition is 'same box, add matching numbers'. It is not matrix multiplication (next page).
Formula
A = (-1)A. Meaning
Where / symbols
- ADDITION AND SCALAR MULTIPLY
- Add (or subtract) entrywise, only if the sizes match. Scalar k times A multiplies every entry by k. Negative: -A = (-1)A. Meaning: addition is 'same box, add matching numbers'. It is not matrix multiplication (next page).
- 1. ADD AND SUBTRACT
- (A+B)_{ij} = a_{ij}+b_{ij}. Commutative: A+B=B+A. Associative: (A+B)+C=A+(B+C). A+O=A.
- 2. SCALAR
- 2A doubles every number. (k+m)A = kA+mA. k(A+B)=kA+kB. Complex scalar: same rule, using complex arithmetic on each entry.
Overview
Add matching boxes:
Add and Subtract
(A+B)ij = aij+bij. Commutative: A+B=B+A. Associative: (A+B)+C=A+(B+C). A+O=A.
Examples [[1,2]]+[[3,4]]=[[4,6]] · Sizes must match · Subtract: minus each pair
Scalar
2A doubles every number. (k+m)A = kA+mA. k(A+B)=kA+kB. Complex scalar: same rule, using complex arithmetic on each entry.
Examples -A flips signs · 0 A = O · Distributive
WHAT — ADDITION AND SCALAR MULTIPLY
Add (or subtract) entrywise, only if the sizes match. Scalar k times A multiplies every entry by k. Negative: -A = (-1)A. Meaning: addition is 'same box, add matching numbers'. It is not matrix multiplication (next page).
RULE — 1. ADD AND SUBTRACT
(A+B)ij = aij+bij. Commutative: A+B=B+A. Associative: (A+B)+C=A+(B+C). A+O=A.
METHOD — 2. SCALAR
2A doubles every number. (k+m)A = kA+mA. k(A+B)=kA+kB. Complex scalar: same rule, using complex arithmetic on each entry.
WORKED EXAMPLE
Worked: 2[[1, -i],[0, 3]] - [[1, i],[2, 1]]. 2A = [[2, -2i],[0, 6]] minus B: [[2-1, -2i-i],[0-2, 6-1]] = [[1, -3i],[-2, 5]]
How to answer a addition and scalar multiply question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- 2[[1, -i],[0, 3]] - [[1, i],[2, 1]].
- 2A = [[2, -2i],[0, 6]]
- minus B: [[2-1, -2i-i],[0-2, 6-1]]
- = [[1, -3i],[-2, 5]]
In simple words
Do not add a 2x2 to a 2x3. Check size before you start.
Key points
- same size to add
- kA multiplies every entry
- entrywise
- every entry
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Addition and Scalar Multiply
Sticky Notes · Quick Revision
Core Idea
- Add (or subtract) entrywise, only if the sizes match.
- Scalar k times A multiplies every entry by k.
- Negative: -A = (-1)A.
1. ADD AND SUBTRACT
- (A+B)ij = aij+bij.
- Commutative: A+B=B+A.
- Associative: (A+B)+C=A+(B+C).
- A+O=A.
2. SCALAR
- 2A doubles every number.
- (k+m)A = kA+mA.
- k(A+B)=kA+kB.
- Complex scalar: same rule, using complex arithmetic on each entry.
Worked Example
- Worked: 2[[1, -i],[0, 3]] - [[1, i],[2, 1]].
- 2A = [[2, -2i],[0, 6]]
- minus B: [[2-1, -2i-i],[0-2, 6-1]]
- = [[1, -3i],[-2, 5]]
Common Misconception
- Do not add a 2x2 to a 2x3. Check size before you start.
FBISE Exam Check
- State the rule for addition and scalar multiply before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Add (or subtract) entrywise, only if the sizes match.
- Scalar k times A multiplies every entry by k.
- Negative: -A = (-1)A.
- Do not add a 2x2 to a 2x3. Check size before you start.
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