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Chapter 5: Q 7E (page 263)

Question:All nonzero symmetric matrices are invertible.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Definition of a symmetric matrix.

Suppose A is a n 脳 nmatrix.

Then the matrix is said to be symmetric matrix, ifAT= A.

A matrix is said to be invertible, if the determinant of the matrix is non-zero.

02

Check whether the given statement is a true or false.

The given statement is all nonzero symmetric matrices are invertible.

For example.

Take,A=1111.

Then,

AT=1000T=1000=A

So,A is a symmetric matrix, but it is not invertible, becausedetA=0.

This means,every symmetric matrix cannot be invertible.

Then the given statement is false.

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Most popular questions from this chapter

Consider an invertible n脳nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof AT.

In Exercises 40 through 46, consider vectors in v鈫�1,v鈫�2,v鈫�3; we are told thatrole="math" localid="1659495854834" v鈫�i,v鈫�j is the entryaij of matrix A.

A=[35115920112049]

46. Find projv(v鈫�3), where V =span role="math" localid="1659495997207" (v鈫�1.v鈫�2). Express your answer as a linear combination ofrole="math" localid="1659496026018" v鈫�1 and v鈫�2.

If A and B are arbitrary n脳nmatrices, which of the matrices in Exercise 21 through 26 must be symmetric?

ATA.

This exercise shows one way to define the quaternions,discovered in 1843 by the Irish mathematician Sir W.R. Hamilton (1805-1865).Consider the set H of all 4脳4matrices M of the form

M=[p鈭�q鈭�r鈭�sqps鈭�rr鈭�spqsr鈭�qp]

where p,q,r,s are arbitrary real numbers.We can write M more sufficiently in partitioned form as

M=(A鈭�BTBAT)

where A and B are rotation-scaling matrices.

a.Show that H is closed under addition:If M and N are in H then so isM+N

M+Nb.Show that H is closed under scalar multiplication .If M is in H and K is an arbitrary scalar then kM is in H.

c.Parts (a) and (b) Show that H is a subspace of the linear space R4脳4 .Find a basis of H and thus determine the dimension of H.

d.Show that H is closed under multiplication If M and N are in H then so is MN.

e.Show that if M is in H,then so is MT.

f.For a matrix M in H compute MTM.

g.Which matrices M in H are invertible.If a matrix M in H is invertible isM鈭�1 necessarily in H as well?

h. If M and N are in H,does the equationMN=NMalways hold?

Using paper and pencil, perform the Gram-Schmidt process on the sequences of vectors given in Exercises 1 through 14.

12.230644213
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