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Chapter 5: Q32E (page 233)

(a) Consider an matrix A such that A螕A=Im. It is necessarily true that? Explain.

(b) Consider an n脳nmatrix A such that ATA=In. Is it necessarily true that AAT=In? Explain.

Short Answer

Expert verified
  1. Not always true for condition (a).
  2. Always true for condition (b).

Step by step solution

01

Determine AAT≠I3.

To obtain AAT鈮�I3consider the matrix below.

A=100100

In the matrix A. performing product,

AAT=I2.

Which shows that AAT鈮�I3

Thus, according to condition (a) it is not necessarily true that AAT=I3.

02

Consider the definition of inverse matrix.

The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity. For a matrix A, its inverse is A-1.

And the formula of inverse matrix is given below.

A-1=1A.AdjA

For example,

A=1-234A=0.40.2-0.30.1

Thus, by consider the definition of inverse matrix above the condition (b) is always true.

Hence, condition (a) is not necessarily true and condition (b) is always true.

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