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Chapter 6: Q17E (page 309)

If twon脳n matrices A and B are similar, then the equationdetA=detB must hold.

Short Answer

Expert verified

Therefore, the given condition is satisfied. 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 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� m by n 鈥� matrix, written 鈥� m脳n.鈥�

02

To check whether the given condition is true or false. 

If A and B are similar, then there exists an invertible matrix T such that,

A=T-1BT

So,

detA=det-1detBdetTdetA=1detTdetBdetTdetA=detB

Therefore, the given condition is satisfied. So, the given statement is true.

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Most popular questions from this chapter

If the determinant of a2脳2 matrix is 4, then the inequality |Av鈫�|鈮�4|v鈫�| must hold for all vectors v鈫� in.

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.

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

2.[123168-2-40]

If is any noninvertible square matrix, then detA=det(rrefA).

If all the entries of a square matrixAare integers and detA=1 , then the entries of matrix A-1 must be integers as well.
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