Transpose of a Matrix
If A = [aij] is mxn matrix, then the matrix of order n x m obtained by interchanging the rows and columns of A is called the transpose of A.It is denoted At or A‘, for example:
Properties of Transpose of a Matrix
• If A‘ and B‘ are tranpose of A and B respectively, then
• (A‘)‘ = A
• (A + B)‘ = A + B, A and B being comparable.
• (KA) = KA, K being any complex number.
• If A and B are of suitable size for AB to exist, then (A B)‘ = B‘ + A‘ (Reversal Law of Transposes)
A square matrix A is called symmetric if A = At, for example:
Thus A is symmetric
A square matrix A is called skew symmetric if A = -At, for example:
A square matrix A is called singular if |A| = 0 , for example if
A square matrix A is called singular if |A| != 0 , for example if
Trace of Matrix
If A is square nxn matrix, its trace, denoted by tr A, is defined to be the sum of the term on the leading diagonal.
tr A = a11 + a22 + —-+ ann
If A is square matrix (aij) and A11, A12 — etc. are the Cofactor of elements a11, a12 — etc. respectively, then matrix [Aij] is called cofactor Matrix of A.
Adjoint of a Matrix
The transpose of cofactor matrix of a given matrix A is called the Adjoint of A or adj A.
Let A = (aij) be a square matrix of order n x n and (cij) is a matrix obtained by replacing each element aij by its corresponding cofactor cij then (cij)t is called the adjoint of A.
adj A = (cij)t
Inverse of a Matrix
If A and B are two square matrices such that AB = BA = 1, then B is called the Inverse of A and vice-versa.
A (adj A)|A| = I, Its behaves as the inverse of A. Hence
A-1 = (adj A)|A| = (adj A)det A
If A is a non-singular square matrix , then
The necessary and sufficient condition for a square matrix to have an inverse is that it should be a non-singular matrix.
• A-1A = AA-1 = 1
• (AB)-1 = B-1A-1
• (A-1)-1 = A
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