184 CHAPTER 8. RANK OF A MATRIX

57. Let A be an n×n matrix and consider the matrices{

I,A,A2, · · · ,An2}. Explain why

there exist scalars, ci not all zero such that

n2

∑i=1

ciAi = 0.

Then argue there exists a polynomial, p(λ ) of the form

λm +dm−1λ

m−1 + · · ·+d1λ +d0

such that p(A) = 0 and if q(λ ) is another polynomial such that q(A) = 0, then q(λ )is of the form p(λ ) l (λ ) for some polynomial, l (λ ) . This extra special polynomial,p(λ ) is called the minimal polynomial. Hint: You might consider an n×n matrixas a vector in Fn2

.

58. Here are some invertible matrices. Write them as a product of elementary matrices.

(a)

 1 2 11 3 01 2 2



(b)

 1 1 −32 3 −71 1 −4



(c)

 2 3 91 2 61 1 2



(d)

 2 4 22 2 21 2 2



59. Here is a matrix:

 1 1 32 1 52 3 7

. Find a product of elementary matrices which,

when this product multiplies the given matrix on the left, the result will be in rowreduced echelon form.

60. Explain why such a sequence of elementary matrices which row reduces a givenmatrix to row reduced echelon form is not unique. That is, tell why you could havetwo different products of elementary matrices which produce the same result.