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In the previous chapter we have studied inverse of a matrix In this section we...

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Algebra

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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 4 5 1 Adjoint of a matrix n Definition 3 The adjoint of a square matrix A a is defined as the transpose of the matrix A where A is the cofactor of the element a Adjoint of the matrix A is denoted by adj A 1 n y Let 88 a11 912 913 A 9 1 922 923 a31 32 33 Hence MATHEMATICS A11 A12 A13 Then adj A Transpose of A21 A22 A23 A31 A32 A33 Rationalised 2023 24 3 Example 12 Find adj A for A 4 Solution We have A 4 A 1 A 3 A 2 2 Let Remark For a square matrix of order 2 given by adj A AA 42 A22 1 adj A A of The adj A can also be obtained by interchanging a a 2 and a212 i e anl a21 911 912 921 a22 where I is the identity matrix of order n Verification A11 A21 A31 A12 A22 A32 A13 A23 A33 a12 Change sign Interchange We state the following theorem without proof Theorem 1 If A be any given square matrix of order n then a11 912 a13 A 921 922 923 then adj A A adj A adj A A A I A21 a a 12 3 and by changing signs Autished All A21 A31 A12 A22 A32

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