Question

- For solving systems, you will be given a simple 2x2 matrix and
the following:

Find ( A -λ I)K = 0; Find K_{1}and K_{2}

Final Equation: X = C_{1}_________________ + C_{2}___________________________

Answer #1

and are solution of the system then ,

, .

So the final equation is because ,

So is a solution of .

Hence Final equation : .

.

**N.B:** To find
and
explictly you need to provide the matrix A .

Find general solutions of the following systems using variation
of parameters.
X ′ =(2x2 matrix) ( 2 2; 3 1 ) X + (column matrix) ( e^−4t; 0
)

Find a particular solution to the following nonhomogeneous
linear systems"
X' = [ 3 -3 ] X + [4]
[ 2 -2 ] [-1]
First is a 2x2 matrix and second is a 1x2 matrix.

Given the following vector X, find a non-zero square matrix A
such that AX=0:
You can resize a matrix (when appropriate) by clicking and dragging
the bottom-right corner of the matrix.
x = [-1]
[10]
[-4]
This is a 3x1 matrix.

Solving Systems of Linear Equations Using Linear
Transformations
In problems 2 and 5 find a basis for the solution set of the
homogeneous linear systems.
2. ?1 + ?2 + ?3 = 0
?1 − ?2 − ?3 = 0
5. ?1 + 2?2 − 2?3 + ?4 = 0
?1 − 2?2 + 2?3 + ?4 = 0.
So I'm in a Linear Algebra class at the moment, and the
professor wants us to work through our homework using...

Which one of the following is the solution to the differential
equation of a 2x1 matrix [x'(t) y'(t)]=the 2x2 matrix [2 3; 3
2][x(t) y(t)] with initial condition of a 2x1 matrix [x(0)
y(0)]=the 2x1 matrix [2 4]?

Solving Systems via the Eigenvalue Method: Find
the general solution to the following system of differential
equations.
x'1 = 3x1 + 4x2
x'2 = 2x1 + x2

Find the fundamental matrix solution for the system x′ = Ax
where matrix A is given. If an initial condition is provided, find
the solution of the initial value problem using the principal
matrix.
A= [ 4 -13 ; 2 -6 ]. , x(o) = [ 2 ; 0 ]

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

Prove the following: Given k x m matrix A, m x n matrix B. Then
rank(A)=m --> rank(AB)=rank(B)

A) Find the inverse of the following square matrix.
I 5 0 I
I 0 10 I
(b) Find the inverse of the following square matrix.
I 4 9 I
I 2 5 I
c) Find the determinant of the following square matrix.
I 5 0 0 I
I 0 10 0 I
I 0 0 4 I
(d) Is the square matrix in (c) invertible? Why or why not?

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