Algebra of Matrices
Any two matrices can be added if they are of the same order and the resulting matrix is of the same order. If two matrices A and B are of the same order, they are said to be conformable for addition.
For example:
Only matrices of the same order can be added or subtracted.
Addition of matrices is commutative as well as associative.
Cancellation laws hold well in case of addition.
The equation A + X = 0 has a unique solution in the set of all m × n matrices.
The matrix obtained by multiplying every element of a matrix A by a scalar λ is called the multiple of A by λ and its denoted by λ A i.e. if A = [a_{ij}] then λA = [λa_{ij}].
For example:
Note: All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalar.
Two matrices can be multiplied only when the number of columns in the first, called the prefactor, is equal to the number of rows in the second, called the postfactor. Such matrices are said to be conformable for multiplication.
where c_{ij} = a_{i1} b_{1j} + a_{i2} b_{2j} +...+ a_{in} b_{nj}= ∑^{n}_{k=1 }a_{ik} b_{ki} i = 1, 2, 3 ......, m and
j = 1, 2, 3 ......, p.
Illustration:
show that AB ≠ BA.
Solution:
Here A.B =
and B.A =
Thus A.B ≠ B.A.
Illustration:
Solution:
We have