Dr. Mark V. Sapir

Class 5

Exercises

Homework due Class 6.

1. Let A be the matrix

[ 3	1 ]
[ 2	1 ]

Find p(A) where p(x) = 2x² - x + 1

2. Show that the matrix

[ a₁₁	0	0	0	...	0
[ 0	a₂₂	0	0	...	0 ]
..............................
[ 0	0	0	0	...	a_nn ]

where a₁₁a₂₂..a_nn does not equal 0 is invertible and find its inverse.

3. Show that if a square matrix A satisfies A² - 3A + I = 0, then A^-1 = 3I- A.

4. Is the sum of two invertible matrices necessarily invertible?

5. Let A and B be square matrices such that AB = 0. Show that if A is invertible, the B = 0.

6. Determine whether the following matrix A is invertible, and if so, find its inverse. [Hint: Solve AX=I by equating corresponding entries on the two sides.]

[ 1	0	1 ]
[ 1	1	0 ]
[ 0	1	1 ]

7. Show that if A is invertible and AB = AC, then B = C.

You may use Maple to solve 1 and 6. Also, you may use problems 3, 5, and 6 to solve problems due Class 7.

Solutions

Class 6

Exercises

I recommend everybody to look at the solutions. In particular, I tried to show what it means to justify every step.

Typical mistakes in solving these homework problems are:

Using a fact, that needs to be proved, in the proof of this fact (for example you need to prove that a matrix is invertible and in the process of proving it you use the inverse of this matrix --- this is illegal).
Using known facts without references (for example you use the fact that -I is an invertible matrix without referring to the theorem which implies that).

Homework due Class 7.

1. (Cayley-Hamilton theorem for 2 by 2 matrices):
Let

A =	[ a	b ]
A =	[ c	d ]

be a square matrix of order 2, trace(A)=a+d, det(A)=ad-bc. Prove that A satisfies the equation

A² - trace(A) A + det(A) I =0.

Here I is the identity matrix:

I =	[ 1	0 ]
I =	[ 0	1 ]

2. Use problem 1 to show that if A is a 2 by 2 matrix and Aⁿ=0 (the zero 2 by 2 matrix) for some natural number n then A²=0. (Hint: Rewrite the equality in Problem 1 in the form:

A²=aA+bI

where a and b are scalars. Deduce, using one of the Problems from the previous assignment that if b is not 0 then A is invertible. Using another Problem from the previous assignment deduce that if A is invertible then Aⁿ cannot be equal to 0 for any n, so b must be 0. Then prove that a=0. In order to do that, multiply the equality A²=aA by A^(n-2). Thus A²=0*A+0=0.)

3. Find a square 3 by 3 matrix A such that A³ is zero but A² is not zero.

4. (Bonus, 20 points). Prove that if A is a 3 by 3 matrix and Aⁿ=0 for some number n then A³=0.

Solutions

Class 7

Exercises

Homework due Class 8.

Prove Cases 2 and 3 in the Theorem about the product EA.
Express the matrix

[ 1 0 1 ]

[ 0 1 1 ]

[ 1 1 0 ]

as a product of elementary matrices.

Bonus problem (10 points). Prove that if A is a square matrix of order n such that AB=BA for every square matrix B of order n then A=xI_n for some number x.

Bonus problem (10 points). Represent the n by n matrix:

[ 2	1	1	1	...	1 ]
[ 1	2	1	1	...	1 ]
[ 1	1	2	1	...	1 ]
........................................
[ 1	1	1	1	...	2 ]

(this matrix has 2 on the diagonal and 1 everywhere else) as a product of elementary matrices.

Solutions

Class 8

Exercises

Homework due Class 9.

Solve the following system of linear equations by inverting the matrix of coefficients:
```
			             x+2y+4z+8t = 1
			            x+3y+9z+27t = 2
			           x+4y+16z+64t = 3
			          x+5y+25z+125t = 4
```
Do not use the "inverse" operation in Maple, use the algorithm described in the notes (or in the html-book).
Let A be an upper triangular matrix of order n with zeroes on the diagonal. Prove that Aⁿ=0. Hint: use Maple to see what happens to such a matrix when we raise it to the second, third, etc. power. Use matrices of order 5. Find a pattern and formulate a conjecture. Then prove your conjecture.
Prove that the symmetric matrix

[ 1 9 ]

[ 9 1 ]

cannot be represented in the form A*A^T where A is a square matrix which has only real (not arbitrary complex) entries. Hint: suppose that this matrix is equal to A*A^T for some matrix

A = [ a b ]

[ c d ]

Comparing the entries of A*A^T and the entries of the original matrix, show that these a,b,c,d must satisfy 4 equations. Then show that this system of equations does not have real solutions (use the fact that |x+y| does not exceed |x|+|y| for all x and y, here |x| is the absolute value of x). This will mean that A does not exist.

Solutions

Class 9

Exercises

Homework due Class 10.

Represent the following matrix as a sum of a symmetric and a skew-symmetric matrices:

[ 1 2 3 ]

[ 2 3 4 ]

[ 4 5 6 ]
Find the signs of the following permutations:
1. (1,2, 4, 3, 5)
2. (5, 4, 3, 2, 1)
3. (6, 5, 4, 3, 2, 1)
4. (n, n-1, ..., 1) for any n (Hint: find the formula for the number of inversions in this permutation. Try several consecutive values of n and check for which of these values the formula gives even/odd numbers. Formulate a conjecture. You may leave it unproved although the proof is not complicated.)

The "proof" homework due Class 12.

Prove Theorem about skew-symmetric matrices.
Find the determinant of the following n by n matrix (it has 0 on the diagonal and 1 everywhere else):

[ 0 1 1 ... 1 1 1 ]

[ 1 0 1 ... 1 1 1 ]

......................................

[ 1 1 1 ... 1 1 0 ]

(you can use the material of Section 2.2 in the book). Hint: use row operations to reduce this matrix to the row-echelon form.

Bonus problem. 10 points. Due Class 12. Prove the third Theorem about symmetric matrices.

Bonus problem. 10 points. Due Class 12. Prove the Theorem about the sign of a permutation.

Solutions

[ 0	1	1	...	1	1	1 ]
[ 1	0	1	...	1	1	1 ]
......................................
[ 1	1	1	...	1	1	0 ]