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Chapter 6: Q44E (page 308)

IfA is an n脳n matrix of rank n-1, what is the rank of adj(A)? See Exercises 42 and 43.

Short Answer

Expert verified

Therefore, the rank of adj(A)is given by,

dimim(adjA)=1

Step by step solution

01

Matrix Definition. 

Matrix is a set of numbers arrangedin rows and columns soas to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an m鈥� by n鈥� matrix, written 鈥�.m脳n鈥�

02

What is the rank of adj(A).

Ifis a non-invertible matrix, then A(adjA)=0..

Therefore, im(adjA)is a subspace of kerA.

Thus,

dimim(adjA)鈮�dimkerAdimim(adjA)=n-dimimAdimim(adjA)=n-(n-1)dimim(adjA)=1

and since the rank of any matrix can't be 0.

We conclude dimim(adjA)=1.

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