solve the homogeneous system $(A-I)x=0$ and use the result to solve $Ax=x$ for $x=[x_1, x_2, x_3]$

Question

5.

solve the homogeneous system $(A-I)x=0$ and use the result to solve $Ax=x$ for $x=[x_1, x_2, x_3]$

Im pretty damn confused about what im supposed to do here, I computed $A-I$ and multiplied it by $x$ column but I just get everything $= 0 $ when I use gausian or any other method. Does that mean the solution is only the $0$ vector?

Answer 1

$A-I$ is an invertible matrix for the given $A$, so yes, $0$-vector is the only solution. In general, the dimension of the null space of a matrix (the set of solutions to $Ax=0$) and its rank adds up to the number of columns of the matrix. This is known as (part of) the fundamental theorem of linear algebra. For the matrix you have ($A-I$), the rank and the number of columns are both 3, so the null space is 0-dimensional, which means it only contains the 0-vector.