Question

Question 2: Let A = 2 −2 4 3 −2 5 −3 3 −4 . a.) Perform elementary row operations to put A in echelon form. b.) Write A as a product of a lower and upper triangular matrix, A = LU. c.) Compute the determinant of L, U, and A.

Answer #1

A triangular matrix is called unit triangular if
it
is square and every main diagonal element is a 1.
(a) If A can be carried by the gaussian algorithm
to
row-echelon form using no row interchanges,
show that A = LU where L is unit lower
triangular and U is upper triangular.
(b) Show that the factorization in (a) is
unique.

(1) Write down a 3 × 3 matrix, call it A, which is not
triangular (upper or lower) with nonzero deter- minant.
(2) By performing one row operation, change your matrix A into a
matrix B which has determinant 4. (If your matrix A already has
determinant 4, change it to one with determinant 5.)
(3) Compute AB and give the determinant of AB.

Finish the following M-file. Run your function on the same
matrix A as given in the lab.
% LU4 - The function in this M-file computes an LU
factorization
% of a 4 x 4 matrix under the assumption that elimination can
be
% performed without row exchanges. % Input: 4 x 4 matrix A;
% Output: lower triangular matrix L and upper triangular matrix
U.
function [L,U] = LU4(A)
L = eye(4);
U = A;
for j= ...
for...

3. Write the matrix in row-echelon form:
1
2
-1
3
3
7
-5
14
-2
-1
-3
8

Let T be an linear transformation from ℝr to ℝs. Let A be the
matrix associated to T.
Fill in the correct answer for each of the following situations
(enter your answers as A, B, or C).
1. Every row in the row-echelon
form of A has a leading entry.
2. Two rows in the row-echelon form of
A do not have leading entries.
3. The row-echelon form of A has a
leading entry in every column.
4. The row-echelon...

A=
2
-3
1
2
0
-1
1
4
5
Find the inverse of A using the method: [A | I ] → [ I | A-1 ].
Set up and then use a calculator (recommended). Express the
elements of A-1 as fractions if they are not already integers. (Use
Math -> Frac if needed.) (8 points)
Begin the LU factorization of A by determining a first
elementary matrix E1 and its inverse E1-1. Identify the associated
row operation. (That...

Q2. Answer this question by hand. Consider the
matrix
matrix A =
2
6
6
7
Determine the eigenvalues and normalized eigenvectors of
A
Compare the trace of A to the sum of the
eigenvalues of A.
Compare the determinant of A to the product of
the eigenvalues of A.
Find
A−1
using elementary row operations. Be sure to indicate the elementary
row operation used at each step of your calculation.
Apply the formula A-1 =
adj(A) = to obtain...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Please give examples of matrices which
(1) is of size 2 × 4, in row echelon form but not reduced row
echelon form, with exactly 6 zero entries.
(2) is of size 5 × 3, in reduced row echelon form with exactly
one zero row.

Let M = ( (−3 1 3 4), (1 2 −1 −2), (−3 8 4 2))
14. (3 points) Let B1 be the basis for M you found by row
reducing M and let B2 be the basis for M you found by row reducing
M Transpose . Find the change of coordinate matrix from B2 to
B1.

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