MathematicsChapter 3: Matrices and Determinants
Matrix Multiplication
AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p).
Learning objectives
- Explain matrix multiplication using the correct definitions, conditions, and notation.
- Apply inner n must match, AB need not equal BA correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p). The (i,j) entry of AB is row i of A dotted with column j of B. AB is usually not BA even when both exist. Multiplication is not entry-wise.
Mathematical form
AB = [[19, 22], [43, 50]]
AB =
Where / symbols
- MATRIX MULTIPLICATION
- AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p). The (i,j) entry of AB is row i of A dotted with column j of B. AB is usually not BA even when both exist. Multiplication is not entry-wise.
- WHEN IT IS DEFINED
- A 1x2 times a 2x1 is 1x1. A 2x3 times a 3x4 is 2x4. If A is 2x3 and B is 2x3, AB is not defined (3 != 2).
- COMPUTE AND ORDER
- For A=[[1,2],[3,4]] and B=[[5,6],[7,8]], (1,1) of AB is 1*5+2*7=19. Full AB=[[19,22],[43,50]]. BA is different: [[23,34],[31,46]].
Overview
Row times column:
When It Is Defined
A 1x2 times a 2x1 is 1x1. A 2x3 times a 3x4 is 2x4. If A is 2x3 and B is 2x3, AB is not defined (3 != 2).
Examples (2x3)(3x4)=2x4 · (1x2)(2x1)=1x1 · Same order is not the rule
Compute and Order
For A=[[1,2],[3,4]] and B=[[5,6],[7,8]], (1,1) of AB is 1*5+2*7=19. Full AB=[[19,22],[43,50]]. BA is different: [[23,34],[31,46]].
Examples 1*5+2*7=19 · AB != BA · Associative: A(BC)=(AB)C
WHAT — MATRIX MULTIPLICATION
AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p). The (i,j) entry of AB is row i of A dotted with column j of B. AB is usually not BA even when both exist. Multiplication is not entry-wise. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Matrix Multiplication 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 WHEN IT IS DEFINED, then COMPUTE AND ORDER. Worked: A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]. Find AB. (1,1): 15 + 27 = 5 + 14 = 19 (1,2): 16 + 28 = 6 + 16 = 22 (2,1): 35 + 47 = 15 + 28 = 43 (2,2): 36 + 48 = 18 + 32 = 50 AB = [[19, 22], [43, 50]] Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]. Find AB.
- (1,1): 1*5 + 2*7 = 5 + 14 = 19
- (1,2): 1*6 + 2*8 = 6 + 16 = 22
- (2,1): 3*5 + 4*7 = 15 + 28 = 43
- (2,2): 3*6 + 4*8 = 18 + 32 = 50
- AB = [[19, 22], [43, 50]]
In simple words
Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
Key points
- inner n must match
- AB need not equal BA
- (m x n)(n x p)
- dot products
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Matrix Multiplication
Sticky Notes · Quick Revision
Core Idea
- AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p). The (i,j) entry of AB is row i of A dotted with column j of B. AB is usually not BA even when both exist. Multiplication is not entry-wise.
WHEN IT IS DEFINED
- A 1x2 times a 2x1 is 1x1. A 2x3 times a 3x4 is 2x4. If A is 2x3 and B is 2x3, AB is not defined (3 != 2).
COMPUTE AND ORDER
- For A=[[1,2],[3,4]] and B=[[5,6],[7,8]], (1,1) of AB is 1*5+2*7=19. Full AB=[[19,22],[43,50]]. BA is different: [[23,34],[31,46]].
Worked Method
- Worked: A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]. Find AB.
- (1,1): 1*5 + 2*7 = 5 + 14 = 19
- (1,2): 1*6 + 2*8 = 6 + 16 = 22
- (2,1): 3*5 + 4*7 = 15 + 28 = 43
- (2,2): 3*6 + 4*8 = 18 + 32 = 50
- AB = [[19, 22], [43, 50]]
Common Misconception
- Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
FBISE Exam Check
- Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
60-second revision
- AB exists when columns of A equal rows of B: (m x n)(n x p) = (m x p).
- WHEN IT IS DEFINED: A 1x2 times a 2x1 is 1x1.
- COMPUTE AND ORDER: For A=[[1,2],[3,4]] and B=[[5,6],[7,8]], (1,1) of AB is 1*5+2*7=19.
- Worked-method reminder: AB = [[19, 22], [43, 50]]
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