midterm1review

midterm1review - 7. Consider the matrix A = 1 2 2 3 1-1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Practice Problems for Midterm 1 1. Consider the following linear system x - y + z = 1 4 x + y + z = 5 2 x +3 y - z = c . (a) Give the augmented matrix of the system and transform it to a reduced row echelon form. (b) Decide how many solutions the system has according to the value of a . (c) In the case that the system has infinitely many solutions, find a general solutions in a parameterized form. 2. Consider the matrix A = 1 2 0 3 2 0 1 0 3 2 0 - 1 2 . (a) Find A - 1 . (b) Find A 3 . 3. Consider two invertible matrices A = 0 0 1 2 1 0 0 0 1 3 0 , B = 1 3 - 1 2 0 1 5 2 0 . (a) Find A - 1 . (b) Suppose that there is another matrix C satisfying BC = A . Find C - 1 . 4. Let A and B be symmetric n × n matrices. Also let C = AB . Show that AB = BA if C is symmetric. 5. Consider n × n invertible matrix A . Prove the following two statements. (a) ( A T ) - 1 = ( A - 1 ) T . (b) A 2 = I if and only if A = A - 1 . 6. Facor the matrix A = 2 0 1 - 4 1 0 6 3 1 into a LU formation.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7. Consider the matrix A = 1 2 2 3 1-1 1-2-2 1 3 1 1 2 1 . Evaluate det( A ), det( A 2 ) and det( A-1 A T ). Answers 1.(a) 1 0 2 5 0 1-3 5 0 0 6 5 1 5 c-3 (b) If c = 3, there are innitely many solution. If c 6 = 3, the system is inconsistent and there is no solution. (c) x =-2 5 + 6 5 y = 3 5 + 1 5 z = . 2. (a) the same as A . (b) the same as A . 3. (a) 0 1 0 0 0 3 2 0 0 . (b) 2 1 15 6 2 6-2 . 6. 1 0 0-2 1 0 3 3 1 2 0 1 0 1 2 0 0-8 7. det( A )=4, det( A 2 )=16 and det( A-1 A T )=1...
View Full Document

Page1 / 2

midterm1review - 7. Consider the matrix A = 1 2 2 3 1-1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online