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Hello.
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In this problem, we have the following matrix.
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We have 2 -0 -0 -0 -negative 2, negative 2, negative 2, negative 2, 1.
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And the target is to find the eigenvalues and eigenvectors.
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So the first part is to get the eigenvalues by getting the determinant of this matrix, which is the same matrix, but subtracting lambda from the diagonal elements.
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So we have the following matrix afterwards.
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Negative lambda, negative 2, negative 2, negative 2, and 1.
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And we need to get the determinant of this matrix.
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So i decided to take this row to make my determinant calculations.
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So from here, taking the first element, we have two.
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Minus lambda multiplied by now we need to get the determinant of this smaller matrix here so we have minus lambda multiplied by one minus lambda minus four closed bracket and then that's a zero so it doesn't count and then we have minus two so we have minus two and we need to get the determinant of this matrix like right now so we have zero multiplied by negative two that's zero and then we have minus minus two minus lambda so that's minus two lambda so that's our determinant and we have it to be equal to zero so after some algebra we find out that reordering this we have negative lambda cubed plus three lambda squared plus six lambda minus eight equal to zero and solving this uh triple third order equation, we find that we have three solutions, whether it's negative 2 or we have it as four or the final eigenvalue is 1.
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In each of these cases, we will have a solution for the eigenvector.
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So let's start by lambda 1 in case of negative 2.
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So we have our matrix 2 minus lambda, which is minus 2.
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So we have it four here and then the rest of the row as it is.
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And then zero.
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And then we have two here.
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And here we have negative two.
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And finally we have negative two.
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Negative two.
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And then one minus minus two.
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That's three.
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And if we multiplied this by x1, x2, x, x, 3, we would get the zero vector.
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So let's solve this equation and see what we can say about it.
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So from the first equation, we have that 4x1 minus 2x3 is equal to 0.
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So from here we have that x1 is simply equal to.
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Take this.
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So half x3.
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So x1 is equal to half x3.
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That's the first piece of info that we can tell from here.
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From the second row equation, we have that 2x2 minus 2 x3 is equal to 0.
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And from here we find that x2 is equal to x3.
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And this is so sufficient to get the equation from here.
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Because actually if we chose like arbitrary value for x2 so like let's say that vector 1 we have x2 is equal to 1 so x3 in that case is equal to 1 and x1 in that case would be equal to half so that's the first vector and of course any scalar of it let's move to the one like right now...