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With t defined by t(x) = Ax, find a vector x whose image under T is b, and determine whether x is unique. Please post step by step :)

A = [1 0 -3

-3 1 6

2 -2 -1],

b = [ -2

3

-1]

User Icespace
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8.9k points

1 Answer

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Sure! Here’s a step-by-step solution to the problem:

First, let’s write the matrix equation that represents the problem: Ax = b, where A is the given matrix, x is the vector we’re trying to find, and b is the given vector.

We can solve this equation by augmenting matrix A with vector b and then performing row operations to put the augmented matrix in reduced row echelon form (RREF).

The augmented matrix is:

[ 1 0 -3 | -2 ]

[-3 1 6 | 3 ]

[ 2 -2 -1 | -1 ]

Copy

We can start by adding 3 times the first row to the second row to eliminate the -3 in the second row:

[ 1 0 -3 | -2 ]

[ 0 1 3 | -3 ]

[ 2 -2 -1 | -1 ]

Copy

Next, we can subtract 2 times the first row from the third row to eliminate the 2 in the third row:

[ 1 0 -3 | -2 ]

[ 0 1 3 | -3 ]

[ 0 -2 5 | 3 ]

Copy

Now we can add twice the second row to the third row to eliminate the -2 in the third row:

[ 1 0 -3 | -2 ]

[ 0 1 3 | -3 ]

[ 0 0 11 | -3 ]

Copy

Finally, we can divide the third row by 11 to get a 1 in the third column:

[1 0 -3 | -2]

[0 1 3 | -3]

[0 0 1 | -3/11]

Copy

Now that we have an RREF matrix, we can solve for x. From the last row of our RREF matrix, we see that x_3 = -3/11. Substituting this value into the second row of our RREF matrix gives us x_2 + (3)(-3/11) = x_2 -9/11 =-3, so x_2 =-24/11. Finally, substituting these values into the first row of our RREF matrix gives us x_1 + (-3)(-3/11) = x_1 +9/11 =-2, so x_1 =-31/11.

Therefore, our solution for x is:

x = [-31/11

-24/11

-3/11]

Copy

Since we were able to solve for a unique solution for x, it means that x is unique.

I hope this helps

User Marianne
by
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