Share

# Operations on Matrices - Multiplication of Matrices

#### description

Non-commutativity of multiplication of matrices, Zero matrix as the product of two non zero matrices

#### notes

The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B. Let A = [a_(ij)] be an m × n matrix and B = [b_(jk)] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p. To get the (i, k)thelement cik of the matrix C, we take the ith row of A and kth column of B, multiply them elementwise and take the sum of all these products. In other words, if A = [a_(ij)]_(m × n), B = [b_(jk)]_(n × p), then the ith row of A is [a_(i1)  a_(i2)  ...  a_("in")] and the k^(th) column of

B is [(b_(1k)) , (b_(2k)), (b_(nk))]
, then c_(ik) =  a_(i1) b_(1k) + a_(i2) b_(2k) + a_(i3) b_(3k) + ... + a_("in") b_(nk)
=$\displaystyle\sum_{j=1}^{n} a_{ij} b_{jk}$.

Non-commutativity of multiplication of matrices:
The below example that even if AB and BA are both defined, it is not necessary that AB = BA.
If A = [(1,0),(0,-1)] and B =[(0,1),(1,0)] ,

then AB [(0,1),(-1,0)]

and BA = [(0,-1),(1,0)].
Clearly AB ≠ BA.
Thus matrix multiplication is not commutative.

Zero matrix as the product of two non zero matrices:
The real numbers a, b if ab = 0, then either a = 0 or b = 0. This need not be true for matrices, we will observe this through an example.
find AB ,if A = [(0,-1),(0,2)] and B = [(3,5),(0,0)]

We have AB = [(0,-1),(0,2)][(3,5),(0,0)]

=[(0,0),(0,0)]
Thus, if the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

### Shaalaa.com

Matrix Multiplication [00:39:48]
S
1
0%