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Chapter 6: Q51E (page 292)

Find the determinant of the(2n)脳(2n)matrix

A=[0lnIn0].

Short Answer

Expert verified

Therefore, the determinant of the given matrix is given by, detA=-1n.

Step by step solution

01

Definition

A determinant is a unique number associated witha square 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

Given

Given matrix,

A=0InIn0

03

To find the determinant

To obtain the identity matrix I2n, we first switch the first and n+1-th row, then we switch the second and n+2-th row, etc.

Lastly, we switch the n-th and 2n -th row.

We had n row interchanges, so detA=-1n.

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Most popular questions from this chapter

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

18.T(f)=2f+3f'fromP2toP2

Even if an n脳n matrix A fails to be invertible, we can define the adjoint adj(A)as in Theorem 6.3.9. The ijthentry of adj(A)is (-1)i+jdet(Aji). For which n脳nmatrices A isadj(A)=0 ? Give your answer in terms of the rank ofA. See Exercise 41.

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

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

21.T(f(t))=f(-t)fromP3toP3

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

7. [0000100012001230123412345]
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