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Chapter 6: Q41E (page 308)

Show that an nxn matrixAhas at least one nonzero minor if (and only if) rank(A)鈮�n-1.

Short Answer

Expert verified

Hence, there exists at least one non-zero minor.

rank(A)鈮�n-1.

Step by step solution

01

Matrix Definition. 

Matrix is a set of numbers arrangedin rows and columns so asto 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 show that at least one nonzero minor. 

If has at least one non-zero minor, it means that the determinant of the matrix calculated for the corresponding minor is different than 0 , which gives us at least n -1 linearly independent rows.

If A has at least n -1 linearly independent rows, then there exists an(n-1)脳n-1 sub matrix of A , and at least one non-zero entry of the row that was left out, hence there exists at least one non-zero minor.

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Most popular questions from this chapter

Explain why any patternPin a matrixA, other than the diagonal pattern, contains at least one entry below the diagonal and at least one entry above the diagonal.

Let Pn be the n脳n matrix whose entries are all ones, except for zeros directly below the main diagonal; for example,

role="math" localid="1659508976827" P5[1111101111101111101111101]

Find the determinant of Pn .

Show that A(adjA)=0=(adjA)Afor all noninvertible n脳nmatrices A. See Exercise 42.

Question:A basisv1鈫�,v2鈫�,v3鈫�of鈩�3is called positively oriented ifv1鈫�encloses an acute angle withv2鈫�脳v3鈫�. Illustrate this definition with a sketch. Show that the basis is positively oriented if (and only if)det[v1鈫�,v2鈫�,v3鈫�]is positive.

Find the determinants of the linear transformations in Exercises 17 through 28.

23.L(A)=ATfrom鈩�n脳nto鈩�n脳n
See all solutions

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