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.