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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.
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Matrix Multiplication
What is it?
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.
In simple words
- 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]]
Important terms
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 15+27=19. Full AB=[[19,22],[43,50]]. BA is different: [[23,34],[31,46]].
What you should learn
- 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.
Examples
- (2x3)(3x4)=2x4
- (1x2)(2x1)=1x1
- Same order is not the rule
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]].
Recommended for Class 10
Full school-level explanation
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.
Row times column:
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.
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).
Remember
- (m x n)(n x p)
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]].
Remember
- dot products
Examples
- 1*5+2*7=19
- AB != BA
- Associative: A(BC)=(AB)C
Remember
- inner n must match
- AB need not equal BA
- [[19, 22], [43, 50]]
- not entry-wise
- (m x n)(n x p)
- dot products
- inner n must match
- AB need not equal BA
How the ideas connect
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]]
Deeper understanding
Go beyond memorising Matrix Multiplication. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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]]
Common mistakes
Exam focus
Core Idea
WHEN IT IS DEFINED
COMPUTE AND ORDER
Worked Method
Exam focus
Quick revision
- inner n must match
- AB need not equal BA
- [[19, 22], [43, 50]]
- not entry-wise
- 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
- 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 !
Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
For long answers on Matrix Multiplication: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- inner n must match
- AB need not equal BA
- [[19, 22], [43, 50]]
- not entry-wise
Important formulas
- inner n must match
- AB need not equal BA
- [[19, 22], [43, 50]]
Common exam traps
- Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
- Same order is not the rule
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
- Mistake
Same order is not the rule
- Mistake
Do not treat WHEN IT IS DEFINED and COMPUTE AND ORDER as the same step; each has a different purpose.
- Mistake
Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
inner n must match
AB need not equal BA
[[19, 22], [43, 50]]
not entry-wise
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
- 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
- Example
- (2x3)(3x4)=2x4
- Critical warning
- Do not multiply matching entries (that would be [[5,12],[21,32]], which is wrong for AB). Use row-dot-column.
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Exam Practice
Define
Q1. Define Matrix Multiplication.
Explain
Q2. Explain: WHEN IT IS DEFINED.
Example
Q3. Give one clear example of Matrix Multiplication and say why it fits.
Mistake
Q4. State one common mistake students make about Matrix Multiplication.
Exam note
Q5. Write a short exam note on Matrix Multiplication: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.