MA-305 Homework 3

Solution

Due at 4:05pm, Thursday, September 18, 1997

You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

1. *Given matrix A as follows, determine if A has an inverse, and find the inverse if it exists. Check your answer by multiplying A and A^-1 to get I.

   	[	1	0	0	0	]
   A=	[	-2	1	0	0	]
   	[	3	2	1	0	]
   	[	1	-2	0	1	]

   Solution:

   	[	1	0	0	0	]
   A-1=	[	2	1	0	0	]	, AA-1 = I.
   	[	-7	-2	1	0	]
   	[	3	2	0	1	]

2. *Assume a is nonzero. Find the inverse of the given matrix.

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

   Solution:

   [	1	0	0	0	]
   [	-a	1	0	0	]
   [	a2	-a	1	0	]
   [	-a3	a2	-a	1	]

3. *Find the LU decomposition for the given matrix. (No interchanges should be required.)

   [	2	-2	6	-4	]
   [	2	-5	2	2	]
   [	-4	1	3	2	]
   [	1	5	1	-2	]

   Solution:

   	[	1	0	0	0	]	[	2	-2	6	-4	]
   LU =	[	1	1	0	0	]	[	0	-3	-4	6	]
   	[	-2	1	1	0	]	[	0	0	19	-12	]
   	[	1/2	-2	-10/19	1	]	[	0	0	0	108/19	]

4. *Use the LU decomposition given for A to solve the equation AX=B, for the given B. Do not find A.

   	[	1				]	[	-3	-1	2	1	]				[	4	]
   A=	[	3	1			]	[		4	-1	2	]	,		B=	[	13	]
   	[	0	2	1		]	[			-2	-1	]				[	6	]
   	[	-2	-1	3	1	]	[				2	]				[	-7	]

   Solution:

   	[	4	]		[	-29/8	]
   Y =	[	1	]	, X =	[	23/8	]	.
   	[	4	]		[	1/2	]
   	[	-10	]		[	-5	]

5. (Also problem 3 of Homework 2)

   [2

   -1]

   **Let

   A=

   [ 2

   1

   -2]

   ,

   B=

   [3

   4]

   [ 3

   2

   5]

   [1

   -2]

   If possible, compute:

   (a) (AB)^T.
   (b) B^TA^T.
   (c) A^TB^T.
   (d) BB^T.
   (e) B^TB.
   

   Solution:

   (a)	[	5	17	]
   	[	6	-5	]

   (b)	[	5	17	]
   	[	6	-5	]

   (c)	[	1	18	-4	]
   	[	0	11	-3	]
   	[	-9	14	-12	]

   (d)	[	5	2	4	]
   	[	2	25	-5	]
   	[	4	-5	5	]

   (e)	[	14	8	]
   	[	8	21	]

*: problems from ``Elmentary Linear Algebra with Applications'' by Hill
**: problem from ``Introductory Linear Algebra with Applications'' by Kolman and Hill