MathematicsChapter 3: Matrices and Determinants
Addition, Subtraction, Scalar
Add or subtract only when both matrices have the same order: combine matching entries.
Learning objectives
- Explain addition, subtraction, scalar using the correct definitions, conditions, and notation.
- Apply same order to add, k multiplies every entry correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Add or subtract only when both matrices have the same order: combine matching entries. Scalar kA multiplies every entry by k. -A is (-1)A. These operations are entry-wise, not row-dot-column (that is multiplication).
Mathematical form
kA = [[3, 12], [6, -3]]
k = 3
kA =
Where / symbols
- ADDITION, SUBTRACTION, SCALAR
- Add or subtract only when both matrices have the same order: combine matching entries. Scalar kA multiplies every entry by k. -A is (-1)A. These operations are entry-wise, not row-dot-column (that is multiplication).
- ADD AND SUBTRACT
- [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]. Subtract matching entries. 4-(-4)=8 when a minus meets a minus.
- SCALAR
- 3[[1,4],[2,-1]] = [[3,12],[6,-3]]. k=-1 flips every sign. This is not the determinant and not the inverse.
Overview
Same shape, then entry by entry:
Add and Subtract
[[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]. Subtract matching entries. 4-(-4)=8 when a minus meets a minus.
Examples 1+5=6 · 4-(-4)=8 · Different orders: cannot add
Scalar
3[[1,4],[2,-1]] = [[3,12],[6,-3]]. k=-1 flips every sign. This is not the determinant and not the inverse.
Examples 3A scales all · -A negates all · Not AB
WHAT — ADDITION, SUBTRACTION, SCALAR
Add or subtract only when both matrices have the same order: combine matching entries. Scalar kA multiplies every entry by k. -A is (-1)A. These operations are entry-wise, not row-dot-column (that is multiplication). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Addition, Subtraction, Scalar matters because matrices organise numbers and give short methods for transformations and simultaneous equations. Correct order, determinant, and multiplication rules prevent most exam errors.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through ADD AND SUBTRACT, then SCALAR. Worked: A = [[9, -1], [4, 2]], B = [[3, 5], [-4, 6]]. Find A - B. A - B = [[9-3, -1-5], [4-(-4), 2-6]] = [[6, -6], [8, -4]] Worked: k = 3, A = [[1, 4], [2, -1]]. Find kA. kA = [[3, 12], [6, -3]] Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- A = [[9, -1], [4, 2]], B = [[3, 5], [-4, 6]]. Find A - B.
- A - B = [[9-3, -1-5], [4-(-4), 2-6]]
- = [[6, -6], [8, -4]]
- k = 3, A = [[1, 4], [2, -1]]. Find kA.
- kA = [[3, 12], [6, -3]]
In simple words
You cannot add a 2x3 to a 3x2. Same number of entries is not enough; the order must match.
Key points
- same order to add
- k multiplies every entry
- matching positions
- every entry times k
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Addition, Subtraction, Scalar
Sticky Notes · Quick Revision
Core Idea
- Add or subtract only when both matrices have the same order: combine matching entries. Scalar kA multiplies every entry by k. -A is (-1)A. These operations are entry-wise, not row-dot-column (that is multiplication).
ADD AND SUBTRACT
- [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]. Subtract matching entries. 4-(-4)=8 when a minus meets a minus.
SCALAR
- 3[[1,4],[2,-1]] = [[3,12],[6,-3]]. k=-1 flips every sign. This is not the determinant and not the inverse.
Worked Method
- Worked: A = [[9, -1], [4, 2]], B = [[3, 5], [-4, 6]]. Find A - B.
- A - B = [[9-3, -1-5], [4-(-4), 2-6]]
- = [[6, -6], [8, -4]]
- Worked: k = 3, A = [[1, 4], [2, -1]]. Find kA.
- kA = [[3, 12], [6, -3]]
Common Misconception
- You cannot add a 2x3 to a 3x2. Same number of entries is not enough; the order must match.
FBISE Exam Check
- You cannot add a 2x3 to a 3x2. Same number of entries is not enough; the order must match.
60-second revision
- Add or subtract only when both matrices have the same order: combine matching entries.
- ADD AND SUBTRACT: [[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]].
- SCALAR: 3[[1,4],[2,-1]] = [[3,12],[6,-3]].
- Worked-method reminder: kA = [[3, 12], [6, -3]]
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