Question

Letv⃗ 1=⎡⎣⎢012⎤⎦⎥,v⃗ 2=⎡⎣⎢1−20⎤⎦⎥,v⃗ 3=⎡⎣⎢30−1⎤⎦⎥be eigenvectors of the matrix A which correspond to the eigenvalues λ1=−3, λ2=2, and λ3=3, respectively, and letx⃗ =⎡⎣⎢−433⎤⎦⎥.Express x⃗ as a linear combination of v⃗ 1, v⃗ 2, and v⃗ 3, and find Ax⃗ .x⃗ = -3 v⃗ 1+ 2 v⃗ 2+ 3 v⃗ 3.Ax⃗ = ⎡⎣⎢⎢⎢⎢⎢⎢ ⎤⎦⎥⎥⎥⎥⎥⎥

Answer 1

The vector x⟨ can be expressed as a linear combination of eigenvectors v⟨1, v⟨2, and v⟨3. To find Ax⟨ we multiply each eigenvector by its corresponding eigenvalue and add the results together, resulting in the vector [9 5 9].

Step-by-step explanation:

We are given the eigenvectors v⟨1, v⟨2, and v⟨3 of a matrix A corresponding to the eigenvalues λ1, λ2, and λ3 respectively. The vector x⟨ can be expressed as a linear combination of these eigenvectors. To find Ax⟨, we can use the definition of eigenvectors, which states that when a matrix A multiplies an eigenvector, the result is the eigenvector scaled by its corresponding eigenvalue.

So, for x⟨ = -3 v⟨1 + 2 v⟨2 + 3 v⟨3, we calculate Ax⟨ by multiplying each eigenvector by its corresponding eigenvalue and adding them together, resulting in:

Ax⟨ = -3(λ1 v⟨1) + 2(λ2 v⟨2) + 3(λ3 v⟨3)
= -3(-3 v⟨1) + 2(2 v⟨2) + 3(3 v⟨3)
= 9 v⟨1 + 4 v⟨2 + 9 v⟨3
= 9⋅0 1 2 + 4⋅1 -2 0 + 9⋅3 0 -1
= 9∗ 0 + 4∗ 1 + 9∗ 3, 9∗ 1 + 4∗ -2 + 9∗ 0, 9∗ 2 + 4∗ 0 + 9∗ -1
= [9 5 9].

Answer 2

What is this????????????????????