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Math 324: Assignment 7 - Determinants and Matrix Operations, Assignments of Mathematics

Instructions for assignment 7 in math 324, which includes evaluating determinants using cofactor expansion and elementary row operations, solving systems of equations using row reduction, inverse matrices, and cramer's rule, and proving properties of determinants for nilpotent, lower triangular, similar, and orthogonal matrices.

Typology: Assignments

Pre 2010

Uploaded on 08/16/2009

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Math 324: Assignment 7
Instructions. The following six exercises plus a written solution to the exercise you select to
present in class are due on Monday, November 1.
1. Evaluate the determinant of
0 1 1
1 2 5
64 3
by:
(a) using the cofactor expansion along the first column;
(b) using the cofactor expansion along the first row.
2. Evaluate the determinant of
1 0 2 3
3 1 1 2
0 4 1 1
2 3 0 1
by
(a) using elementary row operations to make the first entries in rows 2 and 4 both 0, and then
using the cofactor expansion along the first column.
(b) using elementary row operations to make the matrix upper triangular.
3. Do exercise 3. on p. 221.
4. Do exercise 4. on p. 221.
5. Consider the system of equations
3x+y+z= 4
2xy= 12
x+ 2y+z=8
Using MathCAD, solve
this system by
(a) row reducing the agmented matrix;
(b) using the inverse of the coefficient matrix;
(c) using Cramer’s rule.
6. Do exercise 2. on p. 228.
See the reverse side for in-class presentation exercises.
pf2

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Math 324: Assignment 7

Instructions. The following six exercises plus a written solution to the exercise you select to present in class are due on Monday, November 1.

  1. Evaluate the determinant of

 (^) by:

(a) using the cofactor expansion along the first column;

(b) using the cofactor expansion along the first row.

  1. Evaluate the determinant of

 by

(a) using elementary row operations to make the first entries in rows 2 and 4 both 0, and then using the cofactor expansion along the first column.

(b) using elementary row operations to make the matrix upper triangular.

  1. Do exercise 3. on p. 221.
  2. Do exercise 4. on p. 221.
  3. Consider the system of equations

3 x + y + z = 4 − 2 x − y = 12 x + 2 y + z = − 8

Using MathCAD, solve

this system by

(a) row reducing the agmented matrix;

(b) using the inverse of the coefficient matrix;

(c) using Cramer’s rule.

  1. Do exercise 2. on p. 228.

See the reverse side for in-class presentation exercises.

The following exercises are for in-class presentation on Wednesday, November 3.

  1. Let A be an n by n matrix.

(a) A is called nilpotent if Ak^ = O for some k where O is the n by n zero matrix. Prove that |A| = 0 if A is nilpotent.

(b) A is called lower triangular if the all the entries above the main diagonal are 0. Describe |A| in terms of the entries of A.

(c) C is said to be similar to A if there is an invertible matrix B such that C = B−^1 AB. Prove that |A| = |C| if C is similar to A.

(d) The matrix A is said to be orthogonal if AAt^ = I. Prove that |A| = ±1 if A is an orthogonal matrix.

(e) Suppose A is upper triangular. Prove that A is invertible if and only if all of its diagonal entries are nonzero.