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Chapter 6: Q32E (page 309)

There exist real invertible 3脳3 matrices A andSsuch that STAS=-A

Short Answer

Expert verified

Therefore, the given statement is not satisfied.

Step by step solution

01

Matrix Definition

Matrix is a set 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 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

detA=det-A

detA=-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

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.

det(4A)=4detAfor all4脳4 matricesA.

If all the diagonal entries of an n脳nmatrix Aare odd integers and all the other entries are even integers, thenA must be an invertible matrix.

If all the entries of a square matrix are 1 or 0, thenmust be 1, 0, or -1.

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