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Chapter 6: Q42E (page 308)

Even if an n脳n matrix A fails to be invertible, we can define the adjoint adj(A)as in Theorem 6.3.9. The ijthentry of adj(A)is (-1)i+jdet(Aji). For which n脳nmatrices A isadj(A)=0 ? Give your answer in terms of the rank ofA. See Exercise 41.

Short Answer

Expert verified

Therefore,

adjA=0are zero if and only ifrankA鈮�n-2..

Step by step solution

01

Matrix Definition. 

Matrix is a set of numbersarranged 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 an 鈥渕 by n鈥� matrix, written 鈥�m脳n.鈥�

02

To define the adj(A).

We know that,

A matrix A has at least one non-zero minor if and only ifrankA鈮�n-1..

Therefore,

adjA=0ifandonlyifrankA鈮�n-2.

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Most popular questions from this chapter

If all the columns of a square matrixAare unit vectors, then the determinant ofAmust be less than or equal to 1.

For an invertible nxn matrix A, what is the relationship between det (A) and det (A)?

Question:We say that a linear transformation Tfrom N3toN3to preserves orientation if it transforms any positively oriented basis into another positively oriented basis. See Exercise 19. Explain why a linear transformationT(x鈫�)=Ax鈫�preserves orientation if (and only if) detAis positive.

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 Cramer's rule to solve the systems in Exercises 22 through 24.

22.|3x+7y=14x+11y=3|

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