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Chapter 6: Q32 E (page 309)

There exist real invertible 3脳3 matrices A and S such that .

Short Answer

Expert verified

Therefore, the given statement is not satisfied.

Step by step solution

01

Matrix Definition

Matrix is a setof 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 鈥渕 by n鈥� matrix, written 鈥�m脳n .鈥�

02

To check whether the given condition is true or false

On taking determinant

detSdetAdetST=-13detA-detS2detA=detA,

Which is false, as det(S) turns out to be imaginary.

The question is whether there exists an invertible 3脳3matrix A such that A and 2A are similar. If they were similar, it would have to apply

det A = det(-A)

det A = - detA,

Which, since detA鈮�0, is a contradiction. Therefore, the given statement is not satisfied.

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Most popular questions from this chapter

There exists an invertible matrix of the form

[aefjb0g0c0h0d0i0]

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There exists a nonzero4脳4 matrixAsuch thatdetA=detA(4A).

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.

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

9. [1111112222113331114411115]
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