Question

Let A be a 2x2 matrix and suppose that det(A)=3. For each of the following row operations, determine the value of det(B), where B is the matrix obtained by applying that row operation to A.

a) Multiply row 1 by -4

b) Add 4 times row 2 to row 1

c) Interchange rows 2 and 1

Resulting values for det(B):

a) det(B) =

b) det(B) =

c) det(B) =

Answer #1

1. Let A be the 2x2 matrix in M_{2x2}(C), whose first row is
(0,1) and second row is (-1,0).
(a) Show that A is normal.
(b) Find (complex) eigenvalues of A.
(c) Find an orthogonal basis for C^2, which consists of
eigenvectors of A.
(d) Find an orthonormal basis for C^2, which consists of
eigenvectors of A.

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...

Let A equal the 2x2 matrix:
[1 -2]
[2 -1]
and let T=LA R2->R2. (Notice
that this means T(x,y)=(x-2y,2x-y), and that the matrix
representation of T with respect to the standard basis is A.)
a. Find the matrix representation [T]BB
where B={(1,1),(-1,1)}
b. Find an invertible 2x2 matrix Q so that [T]B =
Q-1AQ

Let C = column matrix [1 2 3 1] and D = row matrix[−1 2 1 4] .
Prove that (CD)^10 is the same as C(DC)^9D and do the calculation
by hand.

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...

Let M be an n x n matrix with each entry equal to either 0 or 1.
Let mij denote the entry in row i and column j. A
diagonal entry is one of the form mii for some i.
Swapping rows i and j of the matrix M denotes the following
action: we swap the values mik and mjk for k
= 1,2, ... , n. Swapping two columns is defined analogously.
We say that M is rearrangeable if...

Solve the system
-2x1+4x2+5x3=-22
-4x1+4x2-3x3=-28
4x1-4x2+3x3=30
a)the initial matrix is:
b)First, perform the Row Operation 1/-2R1->R1. The resulting
matrix is:
c)Next perform operations
+4R1+R2->R2
-4R1+R3->R3
The resulting matrix is:
d) Finish simplyfying the augmented mantrix down to reduced row
echelon form. The reduced matrix is:
e) How many solutions does the system have?
f) What are the solutions to the system?
x1 =
x2 =
x3 =

let us create a variable for a row vector a = [1, 4, 1, 3, 2, 5,
0] and calculate the mean value of its elements using the Matlab
function ‘mean’ and store this value in variable aMean. Fig. 1
gives the Matlab code to do this.
a = [1, 4, 1, 3, 2, 5, 0];
aMean = mean(a);
Figure 1: Matlab code – row vector and mean of its elements.
Let us now construct a row vector b that...

1). Show that if AB = I (where I is the identity matrix) then
A is non-singular and B is non-singular (both A and B are nxn
matrices)
2). Given that det(A) = 3 and det(B) = 2, Evaluate (numerical
answer) each of the following or state that it’s not possible to
determine the value.
a) det(A^2)
b) det(A’) (transpose determinant)
c) det(A+B)
d) det(A^-1) (inverse determinant)

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