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Chapter 6: Q53E (page 276)

Find the determinant of the (2n) x (2n)matrixA=[0InIn0]

Short Answer

Expert verified

Therefore, the determinant of (2n( x (2n) is given by,

detA=-1-n.

Step by step solution

01

Definition. 

A determinant is a unique number associatedwith a 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 of the given matrix. 

We have a 2n x 2n block matrix

A=0InIn0

If we switch any two rows of A, the determinant will change its sign. First, we switch the first and (n+1) -th row, then2ndand (n+2) -th row, etc. Lastly, we switch the (n-1) -th and (2n-1) --th row. Now, we've obtained the identity matrixI2n. We had n row interchanges, therefore1=detI2n=-1ndetA.

So,

detA=1-1ndetA=-1n

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Most popular questions from this chapter

A square matrix is called a permutation matrix if each row and each column contains exactly one entry 1, with all other entries being 0 . Examples are In,[010001100], and the matrices considered in Exercises 53 and 56 . What are the possible values of the determinant of a permutation matrix?

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

22.T(f(t))=f(-t)fromPntoPn

If Ais any n x n matrix, thendet(AAT)=det(ATA).

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

8. [0000210003010040010500016]

Show that the function

F[abcdefghj]=bfg

is linear in all three columns and in all three rows. See Example 6. Is F alternating on the columns? See Example 4.
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