Let \(A=\begin{bmatrix}1&0&-2 \\-2&1&6\\3&-2&-5 \end{bmatrix}\) and \(b=\begin{bmatrix}-1 \\7 \\-3 \end{bmatrix}\).

Cassidy Johnson

Cassidy Johnson

Answered question

2022-02-13

Let A=[102216325] and b=[173]. Define a linear transformation T by T(x)=AX. Determine a vector x whose image under T is b. Is the vector x that you found unique or not? Explain your answer.

Answer & Explanation

iluvyou23452nwj

iluvyou23452nwj

Beginner2022-02-14Added 19 answers

So we have to find the vector X that is solution to the following system: Ax+b
Where A is a 3x3 matrix, X is a 3x1 vector like b. This product between a matrix and a vector defines an expanded matrix:
[102|1216|7325|3]
Then the system represented by the expanded matrix is:
x-2z=-1
-2x+y+6z=7
3x-2y-5z=-3
Take first equaiton and add 2z to both side:
x-2z+2z=-1+2z
x=2z-1
Replace x with this expression in the second equation:
-2x+y+6z=7
-2*(2z-1)+y+6z=7
-4z+2+y+6z=7
2+y+2z=7
Substract 2 and 2z from both sides:
2+y+2z=7
2+y+2z-2-2z=7-2-2z
y=5-2z
Replace both x and y with these expression in the third equation:
3x-2y-5z=-3
3*(2z-1)-2*(5-2z)-5z=-3
6z-3-10+4z-5z=-3
5z-13=-3
Add 13 to both sides
5z-13=-3
5z-13+13=-3+13
5z=10
Divide both sides by 5
(5z)/5=10/5
z=2
Use the fact that z is equal to 2 in the expressions for x and y:
x=2z-1
x=2*2-1=3
y=5-2z
y=5-2*2=1
Solution is:
x=3
y=1
z=2
dicky23628u6a

dicky23628u6a

Beginner2022-02-15Added 12 answers

More detailed answer is given below: image
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