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  3. Multiplication of matrices
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Properties of Multiplication of Matrix

Last updated at Dec. 16, 2024 by Teachoo

Let’s look at some properties of multiplication of matrices.

 

1. Commutativity is not true:

  AB ≠ BA

 

2. Zero matrix on multiplication

If AB = O,

then A ≠ O, B ≠ O is possible

 

3. Associative law:

(AB) C = A (BC)

 

4. Distributive law:

A (B + C) = AB + AC

(A + B) C = AC + BC

  •  

5. Multiplicative identity:

For a square matrix A

   AI = IA = A

  where I is the identity matrix of the same order as A.

 

 

Let’s look at them in detail

We used these matrices

 

Commutativity in multiplication is not true

AB ≠ BA

 

Let’s solve them

 

AB

 

BA

 

Since

∴ AB ≠ BA

 

Zero matrix multiplication

 

 

We saw that

So, AB = O

But A ≠ O & B ≠ O

 

 

Therefore,

If two matrices multiply to become zero matrix,

then it is not true that A = O or B = O

 

Note: This is different from numbers

If ab = 0, 
 then either a = 0 or b = 0

But this is not true for matrices

 

Associative law

 

(AB) C = A (BC)

 

Let’s solve this

 

(AB) C

Note: Any matrix multiplied to zero matrix is a zero matrix

  (AB) C = O × C

      = O

 

A (BC)

.

Therefore,

   (AB) C = A (BC)

 

 

Distributive law

 

Distributive law says that -

  • A (B + C) = AB + AC
  • (A + B) C = AC + BC

Let’s prove both of them

 

A (B + C) = AB + AC

 

 

AB + AC

Therefore,

  A (B + C) = AB + AC

Let’s prove the next one

 

(A + B) C = AC + BC

 

 

Therefore,

  (A + B) C = AC + BC

 

Multiplicative Identity

For any square matrix A,

  AI = IA = A

Where I is identity matrix of same order as A

 

Therefore,

  AI = IA = A

 

 

Transcript

Let A = [■8(0&−1@0&2)] B = [■8(3&5@0&0)] C = [■8(3&0@0&4)] AB [■8(0&−1@0&2)] [■8(3&5@0&0)] = [■8(0×3+(−1)×0&0×5+(−1)×0@0×3+2×0&0×5+2×0)] = [■8(0&0@0&0)] BA [■8(3&5@0&0)] [■8(0&−1@0&2)] = [■8(3×0+5×0&3×(−1)+5×2@0×0+0×0&0×(−1)+0×2)] = [■8(0&−3+10@0&0)] = [■8(0&−7@0&0)] Since [■8(0&0@0&0)]≠[■8(0&−7@0&0)] ∴ AB ≠ BA Zero matrix multiplication For A = [■8(0&−1@0&2)], B = [■8(3&5@0&0)] We saw that AB = [■8(0&−1@0&2)][■8(3&5@0&0)] = [■8(0&0@0&0)] = O (AB) C AB = [■8(0&−1@0&2)] [■8(3&5@0&0)] = [■8(0×3+(−1)×0&0×5+(−1)×0@0×3+2×0&0×5+2×0)] = [■8(0&0@0&0)] (AB) C = [■8(0&0@0&0)] [■8(3&0@0&4)] = [■8(0&0@0&0)] A (BC) BC = [■8(3&5@0&0)] [■8(3&0@0&4)] = [■8(3×3+5×0&3×0+5×4@0×3+0×0&0×0+0×4)] = [■8(9&20@0&0)] A (BC) = [■8(0&−1@0&2)][■8(9&20@0&0)] = [■8(0×9+(−1)×0&0×20+(−1)×0@0×9+2×0&0×20+2×0)] = [■8(0&0@0&0)] Therefore, (AB) C = A (BC) A (B + C) B + C = [■8(3&5@0&0)] [■8(3&0@0&4)] = [■8(3+3&5+0@0+0&0+4)] = [■8(9&5@0&4)] A (B + C) = [■8(0&−1@0&2)][■8(9&5@0&4)] = [■8(0×9+(−1)×0&0×5+(−1)×4@0×9+2×0&0×5+2×4)] = [■8(0&−4@0&8)] AB + AC AB = [■8(0&−1@0&2)][■8(3&5@0&0)] = [■8(0×3+(−1)×0&0×5+(−1)×0@0×3+2×0&0×5+2×0)] = [■8(0&0@0&0)] AC = [■8(0&−1@0&2)][■8(3&0@0&4)] = [■8(0×3+(−1)×0&0×0+(−1)×4@0×3+2×0&0×0+2+4)] = [■8(0&−4@0&8)] AB + AC = [■8(0&0@0&0)] + [■8(0&−4@0&8)] = [■8(0&−4@0&8)] Therefore, A (B + C) = AB + AC Let’s prove the next one (A + B) C A + B = [■8(0&−1@0&2)] + [■8(3&5@0&0)] = [■8(0+3&(−1)+5@0+0&2+0)] = [■8(3&4@0&2)] (A + B) C = [■8(3&4@0&2)][■8(3&0@0&4)] = [■8(3×3+4×0&3×0+4×4@0×3+2×0&0×0+2×4)] = [■8(9&16@0&8)] AC + BC AC = [■8(0&−1@0&2)] [■8(3&0@0&4)] = [■8(0×3+(−1)×0&0×0+(−1)×4@0×3+2×0&0×0+2×4)] = [■8(0&−4@0&8)] BC = [■8(3&5@0&0)][■8(3&0@0&4)] = [■8(3×3+5×0&3×0+5×4@0×3+0×0&0×0+0×4)] = [■8(9&20@0&0)] AC + BC = [■8(0&−4@0&8)]+[■8(9&20@0&0)] = [■8(0+9&(−4)+20@0+0&8+0)] = [■8(9&16@0&8)] Therefore, (A + B) C = AC + BC For A = [■8(0&−1@0&2)] I = [■8(1&0@0&1)] AI = [■8(0&−1@0&2)] [■8(1&0@0&1)] = [■8(0×1+(−1)×0&0×0+(−1)×1@0×1+2×0&0×0+2×1)] = [■8(0&−1@0&2)] = A IA = [■8(1&0@0&1)][■8(0&−1@0&2)] = [■8(1×0+0×0&0×(−1)+0×2@0×0+1×0&0×(−1)+1×2)] = [■8(0&−1@0&2)] = A Note: For a 3 × 3 matrix, I will be a 3 × 3 matrix Example – For A = [■8(9&5&2@1&8&5@3&1&6)], I = [■8(1&0&0@0&1&0@0&0&1)]
  1. Chapter 3 Class 12 Matrices
  2. Concept wise

    3. Multiplication of matrices

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Statement questions - Multiplication of matrices →
Chapter 3 Class 12 Matrices
Concept wise

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo