. Using Cofactors of elements of second row, evaluate ∆ = . . Using Cofactors of elements of third column, evaluate ∆ = yz zx xy .
. If ∆ = and A ij is Cofactors of a ij , then value of ∆ is given by (A) a A + a A + a A (B) a A + a A + a A (C) a A + a A + a A (D) a A + a A + a A . Adjoint and Inverse of a Matrix In the previous chapter, we have studied inverse of a matrix. In this section, we shall discuss the condition for existence of inverse of a matrix.
To find inverse of a matrix A, i.e., A – we shall first define adjoint of a matrix. . . Adjoint of a matrix Definition The adjoint of a square matrix A = [ a ij ] n × n is defined as the transpose of the matrix [A ij ] n × n , where A ij is the cofactor of the element a ij .
Adjoint of the matrix A is denoted by adj A. Let A = Then A =Transposeof A adj = A Example Find A for A = adj Solution We have A = , A = – , A = – , A = Hence adj A = – Remark For a square matrix of order , given by A = The adj A can also be obtained by interchanging a and a and by changing signs of a and a , i.e., We state the following theorem without proof. Theorem If A be any given square matrix of order n , then A( adj A) = ( adj A) A = A I , where I is the identity matrix of order n Verification Let A = , then adj A =