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Chapter 6: Q.6.1-45E (page 276)

If is an invertible n脳nmatrix, then must commute with its adjoint, adj(A) .

Short Answer

Expert verified

A路adjA=adjA路A

So, the given statement is true.

Step by step solution

01

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as 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 a 鈥� m by n鈥� matrix, written 鈥� m脳n鈥�.

02

To check whether the given condition is true or false

Ifis an invertible matrix, then

A-1=1detAadjA.

Then,

A路adjA=A路(detA)A-1A路adjA=(detA)AA-1A路adjA=(detA)A路adjA=(detA)A-1A

Therefore,

A路adjA=adjA路A

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Most popular questions from this chapter

In Exercises 62 through 64, consider a function D from 鈩�2脳2 to鈩� that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume thatD(l2)=1.


62. Show thatD(A)=0for any2脳2matrix whose two columns are equal.

Consider two distinct real numbers, a and b. We define the function

f(t)=det[111abta2b2t2]

a. Show thatf(t) is a quadratic function. What is the coefficient oft2?
b. Explain whyf(a)=f(b)=0. Conclude thatf(t)=k(t-a)(t-b), for some constant k. Find k, using your work in part (a).
c. For which values of tis the matrix invertible?

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

8. [0000210003010040010500016]

For two invertible nxnmatrices A and B , what is the relationship between adj(A),adj(B),andadj(AB)?

Which of the following functions F of A=[abcd] are linear in both columns? Which are linear in both rows? Which are alternating on the columns?
role="math" localid="1659502796300" a.F(A)=bcb.F(A)=cdc.F(A)=acd.F(A)=bc-ade.F(A)=c
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