## Inverse of a Matrix

DEFINITION 1.2.13 (Inverse of a Matrix)   Let be a square matrix of order
1. A square matrix is said to be a LEFT INVERSE of if
2. A square matrix is called a RIGHT INVERSE of if
3. A matrix is said to be INVERTIBLE (or is said to have an INVERSE) if there exists a matrix such that

LEMMA 1.2.14   Let be an matrix. Suppose that there exist matrices and such that and then

Proof. Note that

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Remark 1.2.15
1. From the above lemma, we observe that if a matrix is invertible, then the inverse is unique.
2. As the inverse of a matrix is unique, we denote it by That is,

THEOREM 1.2.16   Let and be two matrices with inverses and respectively. Then

Proof. Proof of Part 1.
By definition Hence, if we denote by then we get Thus, the definition, implies or equivalently

Proof of Part 2.
Verify that

Proof of Part 3.
We know Taking transpose, we get

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EXERCISE 1.2.17
1. Let be invertible matrices. Prove that the product is also an invertible matrix.
2. Let be an inveritble matrix. Then prove that cannot have a row or column consisting of only zeros.
3. Let be an invertible matrix and let be a nonzero real number. Then determine the inverse of the matrix .

A K Lal 2007-09-12