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Chapter 6: Q39E (page 307)

If A and Bare invertible n脳nmatrices, and if Ais similar toB, isadj(A) necessarily similar to adj(B)?

Short Answer

Expert verified

Yes, A is similar to B and adjAis similar toadjB .

Step by step solution

01

Matrix Definition. 

Matrix is a set of numbersarranged in 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

If A is B similar to , isadj(A)  necessarily similar to adj(B)

If A is similar to B, then there exists an invertiblen脳nmatrix T

Such that

T-1AT=Badj(T-1AT)=adjB.

Since is similar to , they have the same determinant.

So,

1detTdetAdetTT-1AT-1=(detB)B-1(T-1)-1(adjA)T-1=adjB.

.

Therefore, adjAis similar to adjB.

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