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A matrix built from its products with given vectors: number of solutions of (A - kI)X = b

The same question, asked 2 times: 25 Jun 2022*, 31 Jan 2024. Every wording below, oldest first.

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  1. Question 1

    25 Jun 2022

    65
    Let AA be a 3×33 \times 3 real matrix such that A(110)=(110);A(101)=(−101)A\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix};A\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}=\begin{pmatrix} - 1 \\ 0 \\ 1 \end{pmatrix} and A(001)=(112)A\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}=\begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}. If X=(x1,x2,x3)TX=\left( x_{1},x_{2},x_{3} \right)^{T} and II is an identity matrix of order 3 , then the system (A−2I)X=(411)(A- 2I)X=\begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} has

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  2. Question 2

    31 Jan 2024

    169
    Let AA be a 33 real matrix such that A(101)=2(101),A(−101)=4(−101),A(010)=2(010)A\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}= 2\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix},A\begin{pmatrix} - 1 \\ 0 \\ 1 \end{pmatrix}= 4\begin{pmatrix} - 1 \\ 0 \\ 1 \end{pmatrix},A\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}= 2\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}. Then, the system (A−3I)(xyz)=(123)(A- 3I)\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} has

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