Equality of Two Matrices
Two matrices
A = [aij ] and B = [bij ] are said to be equal if,
- They are of same size.
- The elements in the corresponding places of two matrices are the same i.e., aij = bij for each pair of subscripts i and j.
Addition of Matrices
Two matrices A and B are compatible for addition only if they both have exactly the same size say m × n. Then their sum is defined to be the matrix of the type m × n obtained by adding corresponding elements of A and B. Thus if, A = [aij ]m × n & B = [bij ]m × n then A + B = [aij + bij ]m × n.
Substraction of Two Matrices
If A and B are two m × n matrices, then we define, A – B = A + (–B).
Thus the difference A – B is obtained by subtracting from each element of A corresponding elements of B.
Multiplication of a Matrix by a Scalar
Let A be any m × n matrix and k be any real number called scalar. The m × n matrix obtained by multiplying every element of the matrix A by k is called scalar multiple of A by k and is denoted by kA.
⇒ If A = [aij ]m × n then Ak = kA =[kA]m × n.
Multiplication of Two Matrices
Let A = [aij ]m × n; B = [bjk]n × p be two matrices such that the number of columns in A is equal to the number
of rows in B
Trace of a Matrix
Let A be a square matrix of order n. The sum of the elements lying along principal diagonal is called the trace of A denoted by Tr(A).
Transpose of a Matrix
Let A = [aij ]m × n. Then the n × m matrix obtained from A by changing its rows into columns and its columns into rows is called the transpose of A and is denoted by A′ or AT.
Conjugate of a Matrix
The matrix obtained from given matrix A on replacing its elements by the corresponding conjugate complex numbers is called the conjugate of A and is denoted by A.
Transposed Conjugate of Matrix
The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by Aθ or A* or ( )T A . It is also called conjugate transpose of A. This is also known as tranjugate of A
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