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| Multiplication of Matrices |
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| Let us consider the sales done by a school canteen for two successive days. |
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| Monday 25 cokes 12 cakes |
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| Tuesday 40 cokes 7 cakes |
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| Each coke costs $ 5 and cake costs $ 10. |
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| The sales can be expressed as a 2 x 2 matrix and the price as a 2 x 1 matrix as given below: |
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| The amount collected for two days can be calculated as shown below: |
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| We have shown the product of two matrices with the help of the above example. |
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| A is a 2 x 2 matrix and B is a 2 x 1 matrix. The resultant matrix is 2 x 1. |
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| Two matrices A and B can be multiplied if the number of columns of A is equal to the number of rows of B. The resultant matrix will be of the order of (number of rows of A x number of columns of B). |
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| If A be a matrix of the order m x n and B be a matrix of the order n x q, then A and B can be multiplied and the product will be a matrix of order m x q. |
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then  |
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| Elements of rows of matrix A are multiplied by the corresponding elements of columns of B and we get AB. |
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| = [5 x 3 + 6 x 2] = [27]1 x 1 |
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 is called unit matrix or identity matrix. |
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| A x I = A = I x A. |
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If  . Find (i) A(BC), (ii)(AB)C. |
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| Is A(BC) = (AB)C? Does it possess associate property? |
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 (AB) C = A(BC) |
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 It has associative property. |
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| (1) A x I = I x A = A where I denotes a unit matrix of suitable order. |
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| Matrix I possesses identity property of multiplication, |
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 I is called a unit matrix or identity matrix. |
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(2)  , it does not have commutative property. |
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| (3) A(B + C) = AB + AC (Distributive property) |
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| (4) A(BC) = (AB)C (Associative property) |
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(5) If AB = AC and  then A may or may not be zero. |
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| (6) We write kA and not Ak where k is scalar multiple, |
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| Example: |
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If  is multiplied by 4, we write  and not  |
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