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Chapter 6: Q55E (page 292)

For a fixed positive integer n , letD be a function which assigns to any n脳nmatrixA a number D(A)such that
a. D is linear in the rows (see Theorem 6.2.2),
b. D(B)=-D(A) ifB is obtained fromA by a row swap, and
c. .D(In)=1
Show thatD(A)=det(A) for alln脳n matricesA .

Short Answer

Expert verified

Therefore,

D(A)=detA,鈭赌A鈭�鈩�n脳n.

Step by step solution

01

Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

To show  D(A)=det(A)

For an arbitraryn脳nmatrixA,

Let,

E=rrefA.

We know that Eis an upper triangular matrix, and if there were k row interchanges to obtain E, and from the properties of D,

We haveD(E)=e1,1路e2,2路鈥�路en,n=detE=(-1)kdetA.

On the other hand, using these same properties of D, we can see that

D(E)=D(rrefA)=(-1)kD(A)

Thus,

D(A)=detA,鈭赌A鈭�鈩�n脳n.

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Most popular questions from this chapter

If Ais an invertible n脳n matrix, thendet(AT) must equaldet(A-1).

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

The determinant of any diagonaln脳nmatrix is the product of its diagonal entries.

For which angle(s) 胃 can you find three distinct unit vectors in鈩�2 such that the angle between any two of them is胃 ? Draw a sketch.

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|ax-by=1bx+ay=0|.

What are the signs of the solutions x and y? How does x change as bincreases?
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