DEFINITION 2.6.9 (Minor, Cofactor of a Matrix)   The number is called the minor of . We write The cofactor of denoted is the number

DEFINITION 2.6.10 (Adjoint of a Matrix)   Let be an matrix. The matrix with for is called the Adjoint of denoted

EXAMPLE 2.6.11   Let Then
as and so on.

THEOREM 2.6.12   Let be an matrix. Then
1. for
2. for and
3. Thus,

 (2.6.2)

Proof. Let be a square matrix with
• the row of as the row of
• the other rows of are the same as that of
By the construction of two rows ( and ) are equal. By Part 5 of Lemma 2.6.6, By construction again, for Thus, by Remark 2.6.7, we have

Now,

Thus, Since, Therefore, has a right inverse. Hence, by Theorem 2.5.9 has an inverse and

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EXAMPLE 2.6.13   Let Then

and By Theorem 2.6.12.3,

The next corollary is an easy consequence of Theorem 2.6.12 (recall Theorem 2.5.9).

COROLLARY 2.6.14   If is a non-singular matrix, then
and

THEOREM 2.6.15   Let and be square matrices of order Then

Proof. Step 1. Let
This means, is invertible. Therefore, either is an elementary matrix or is a product of elementary matrices (see Theorem 2.5.8). So, let be elementary matrices such that Then, by using Parts 1, 2 and 4 of Lemma 2.6.6 repeatedly, we get

Thus, we get the required result in case is non-singular.

Step 2. Suppose
Then is not invertible. Hence, there exists an invertible matrix such that where So, and therefore

Thus, the proof of the theorem is complete. height6pt width 6pt depth 0pt

COROLLARY 2.6.16   Let be a square matrix. Then is non-singular if and only if has an inverse.

Proof. Suppose is non-singular. Then and therefore, Thus, has an inverse.

Suppose has an inverse. Then there exists a matrix such that Taking determinant of both sides, we get

This implies that Thus, is non-singular. height6pt width 6pt depth 0pt

THEOREM 2.6.17   Let be a square matrix. Then

Proof. If is a non-singular Corollary 2.6.14 gives

If is singular, then Hence, by Corollary 2.6.16, doesn't have an inverse. Therefore, also doesn't have an inverse (for if has an inverse then Thus again by Corollary 2.6.16, Therefore, we again have

Hence, we have height6pt width 6pt depth 0pt

A K Lal 2007-09-12