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Chapter 3: Q36E (page 144)

Can you find a 3脳3 matrix Asuch that im(A)=ker(A)? Explain.

Short Answer

Expert verified

We cannot find the3脳3 matrixA such that im(A)=ker(A).

Step by step solution

01

To mention given data and recall theorem 3.3.7

Let Abe a 3脳3matrix.

We have given that im(A)=ker(A).

Theorem 3.3.7, is stated as follows:

For anyn脳m matrix A, the equation,

dim(ker(A))+dim(im(A))=m.

Here, in this casen=m=3 .

02

To find the dimension of the space

By Theorem 3.3.7, we have,

m=dim(ker(A))+dim(im(A))鈬�3=dim(im(A))+dim(im(A))鈬�3=2dim(im(A))鈬�dim(im(A))=32

This is impossible.

Hence, we cannot find the3脳3 matrix Asuch that im(A)=ker(A).

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