ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A Q.NO. 7

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

ISS PREVIOUS YEAR 2016 PAPER-2 SOLUTION SET-A

Generalized Inverse of a Matrix – Solved Question

Question

Which of the following statements are correct about the generalized inverse of a matrix A?

1. A generalized inverse of a matrix is defined as

   
   

   A A⁻ A = A

2. A generalized inverse is not unique.

3. A matrix to have a generalized inverse must be non-singular.

4. A matrix to have a generalized inverse may be either a square matrix or a rectangular matrix.

Select the correct answer using the code given below:

(a) 1, 2 and 3(b) 1, 2 and 4(c) 2, 3 and 4(d) 1, 3 and 4

Step-by-Step Solution

Statement 1

A matrix A⁻ is called a generalized inverse (g-inverse) of matrix A if it satisfies the condition:

A A⁻ A = A

This is the basic definition of a generalized inverse in linear algebra.

Therefore,

Statement 1 is correct.

Statement 2

The generalized inverse of a matrix is not unique.

For a given matrix A, there may exist multiple matrices that satisfy the condition A A⁻ A = A.

Therefore,

Statement 2 is correct.

Statement 3

The statement says that a matrix must be non-singular to have a generalized inverse.

This is incorrect.

Even singular matrices can have a generalized inverse. The concept of generalized inverse was introduced specifically to deal with matrices that may not have a regular inverse.

Therefore,

Statement 3 is incorrect.

Statement 4

A generalized inverse can exist for:

• Square matrices

• Rectangular matrices

Unlike the usual inverse (which only exists for non-singular square matrices), a generalized inverse can also be defined for rectangular matrices.

Therefore,

Statement 4 is correct.

Correct Statements

The correct statements are:

1, 2 and 4

Final Answer

Option (b) 1, 2 and 4

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