Dr. Mark V. Sapir

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

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1. Evaluate the determinants of the following matrices using only the definition of determinants:
   1. 
      

      [ 1	2	0	0 ]
      [ 0	1	0	0 ]
      [ 1	0	0	1 ]
      [ 0	0	1	0 ]

   2. 
      

      [ 1	2	3	4 ]
      [ 0	1	2	3 ]
      [ 0	0	1	2 ]
      [ 1	0	0	1 ]

2. Given that the determinant of the following matrix is equal to 5,

   [ a	b	c	d ]
   [ k	l	m	n ]
   [ p	q	r	s ]
   [ x	y	z	t ]

   find the determinant of the following matrix
   

   [ 3a+p	3b+q	3c+r	3d+s ]
   [ k-x	l-y	m-z	n-t ]
   [ x	y	z	t ]
   [ p	q	r	s ]

   (Use the Theorems about determinants.)

Solutions

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Solutions

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Class 12

Homework due Class 13.

1. Compute the determinant of the following matrix using cofactor expansion along the third row:
   

   [ 2	-3	4	1 ]
   [ 4	-2	3	3 ]
   [ a	b	c	d ]
   [ 3	-1	4	3 ]

2. Compute the determinant of the following n by n matrix:
   

   [ x	y	0	0	...	0	0 ]
   [ 0	x	y	0	...	0	0 ]
   [ 0	0	x	y	...	0	0 ]
   ...........................................
   [ 0	0	0	0	...	x	y ]
   [ y	0	0	0	...	0	x ]

3. Compute the determinant of the following n by n matrix by using row operations:

   [ 1	2	3	...	n ]
   [ -1	0	3	...	n ]
   [ -1	-2	0	...	n ]
   ......................................
   [ -1	-2	-3	...	n ]

"Proof" homework. Homework due Class 15.

1. Prove that a square matrix A is a zero divisor that is AB=0 for some non-zero square matrix B if and only if det(A)=0
2. Prove that if all entries of a square matrix A are integers and det(A)=1 then all entries of A^-1 are also integers.

Bonus problems. Due Class 15.

1. (7 points) Prove part f of the first theorem about determinants.
2. (7 points) Prove part c of the second theorem about determinants.
3. (7 points) Prove that the determinant of the following Vandermonde matrix:

   [ 1	x1	x12	...	x1n-1 ]
   [ 1	x2	x22	...	x2n-1 ]
   ......................................
   [ 1	xn	xn2	...	xnn-1 ]

   is equal to the product of all differences x_i-x_j where i> j, i and j are from 1 to n. For example if n=2 then the determinant is equal to x₂-x₁, if n=3 then the determinant is equal to (x₂-x₁)(x₃-x₁)(x₃-x₂).

Solutions

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Homework due Class 14.

1. Solve the following system of equations using Cramer's rule:

   		             4x + 5y + 6z      = 6
   		                  6y + 3z + 2t = 10
   		             3x + 2y +  z -  t = 1
   		              x +            t = 2
   

2. Find all values of parameter a such that in the solution of the following system of equations, we have x>y (use Cramer's rule):

   		                      2x+3y+5z+t=a
   		                      3x- y- z-t=2a
   		                       x +   z+t=3a
   		                       x+ y+ z+t=5a
   

Solutions

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