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Chapter 6: Q37E (page 306)

If an n脳n matrixAis invertible, then there must be an (n-1)脳(n-1)sub matrix of(obtained by deleting a row and a column of) that is invertible as well.

Short Answer

Expert verified

Therefore, the sub matrix is invertible. 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 rectangular array.

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 by n鈥� matrix, written 鈥渕脳n.鈥�

02

To check whether the given condition is true or false

If A is an invertible matrix, thendetA鈮�0 .

Thus, if we use the Laplace expansion along the first row, we can see that at least one of the addends must be different than 0.

Therefore, this must also apply to at least one n-1脳n-1sub matrix.

So that sub matrix is invertible.

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Most popular questions from this chapter

Question: Arguing geometrically, determine whether the following orthogonal transformations from鈩�3to鈩�3preserve or reverse orientation. See Exercise 20.
a. Reflection about a plane

b. Reflection about a line
c. Reflection about the origin

If all the diagonal entries of an n脳nmatrixA are even integers and all the other entries are odd integers, then Amust be an invertible matrix.

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

24. |2x+3y=84y+5z=36x+7z=-1|

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

22.T(f(t))=f(-t)fromPntoPn

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