210 CHAPTER 11. MATRICES AND THE INNER PRODUCT

(d)

 1 0 01 1 11 0 2



(e)

 2 0 11 1 1−1 0 0



(f)

 6 −2 33 0 2−5 2 −2

4. Suppose you have p(λ ) is the minimum polynomial for a square n× n matrix A.

Show that this matrix is invertible if and only if the constant term of the minimumpolynomial is non zero. In this case, give a formula for A−1 in terms of powers of A.Say

p(λ ) = λn +an−1λ

n−1 + · · ·+a1λ +a0

Thus you need explain why a0 ̸= 0 if A−1 exists and then find a formula for A−1

when this is the case.

5. Find least squares solutions to the following systems of equations.

(a)

 1 2−1 12 1

( xy

)=

 111

(b)

(1 12 2

)(xy

)=

(11

)

(c)

 1 0 11 1 02 1 1

 x

yz

=

 102

6. Here are some matrices. Label according to whether they are symmetric, skew sym-

metric, or orthogonal. If the matrix is orthogonal, determine whether it is proper orimproper.

(a)

 1 0 00 1/

√2 −1/

√2

0 1/√

2 1/√

2

(b)

 1 2 −32 1 4−3 4 7

 (c)

 0 −2 −32 0 −43 4 0

7. Show that every real matrix may be written as the sum of a skew symmetric and a

symmetric matrix. Hint: If A is an n× n matrix, show that B ≡ 12

(A−AT

)is skew

symmetric.

8. Let x be a vector in Rn and consider the matrix I− 2xxT

||x||2. Show this matrix is both

symmetric and orthogonal.