Multiplication of Matrices

Let us consider the sales done by a school canteen for two successive days.

Monday 25 cokes 12 cakes

Tuesday 40 cokes 7 cakes

Each coke costs $ 5 and cake costs $ 10.

The sales can be expressed as a 2 x 2 matrix and the price as a 2 x 1 matrix as given below:

The amount collected for two days can be calculated as shown below:

We have shown the product of two matrices with the help of the above example.

A is a 2 x 2 matrix and B is a 2 x 1 matrix. The resultant matrix is 2 x 1.

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).

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.

then

Elements of rows of matrix A are multiplied by the corresponding elements of columns of B and we get AB.

= [5 x 3 + 6 x 2] = [27]_{1 x 1}

is called unit matrix or identity matrix.

A x I = A = I x A.

If . Find (i) A(BC), (ii)(AB)C.

Is A(BC) = (AB)C? Does it possess associate property?

(AB) C = A(BC)

It has associative property.

Some properties of Multiplication of Matrices

(1) A x I = I x A = A where I denotes a unit matrix of suitable order.

Matrix I possesses identity property of multiplication,

I is called a unit matrix or identity matrix.

(2) , it does not have commutative property.

(3) A(B + C) = AB + AC (Distributive property)

(4) A(BC) = (AB)C (Associative property)

(5) If AB = AC and then A may or may not be zero.

(6) We write kA and not Ak where k is scalar multiple,

Example:

If is multiplied by 4, we write and not