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Chapter 8: Q17E (page 392)

Use the approach of Exercise 16to find the determinant of the nxn matrix Bthat has p鈥檚 on the diagonal and q鈥檚 elsewhere:

B=[pq鈥qqp鈥q鈰鈰鈰鈰qq鈥p]

Short Answer

Expert verified

The determinant of the matrixB=pq鈥qqp鈥q鈰鈰鈰鈰qq鈥p is detB=p-qn-1qn+p-q.

Step by step solution

01

 Step 1: Define symmetric matrix

  • In linear algebra, a symmetric matrix is a square matrix that does not change when its transpose is calculated.
  • A symmetric matrix is defined as one whose transpose is identical to the matrix itself.
  • A square matrix of sizenxnis symmetric ifBT=B .
02

Find the determinant of the given matrix

Consider A to be an all-ones nxn matrix,

A=11鈥1111鈥11鈰鈰鈰鈰鈰鈰鈰鈰鈰鈰1鈥鈥鈥1

From the previous exercise we found the eigenvalues of位=0,位=nwith multiplicities n-1,1. The given matrix can be rewritten as,

pq鈥qqqp鈥qq鈰鈰鈰鈰鈰鈰鈰鈰鈰鈰q鈥鈥鈥p=q11鈥1111鈥11鈰鈰鈰鈰鈰鈰鈰鈰鈰鈰1鈥鈥鈥1+p-q100鈥0010鈥0001鈥0鈰鈰鈰鈰鈰000鈥1

Theeigenvaluesare位=p-qand位=qn+p-qwithmultiplicityn-1,1.ThedeterminantofthegivenmatrixisdetB=p-qn-1qn+p-q.

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