Let A be a 33 real matrix such that Apmatrix110pmatrix=pmatrix111pmatrix, Apmatrix101pmatr…

Question

Let $A$ be a $3	imes3$ real matrix such that $A\begin{pmatrix}1\1\0\end{pmatrix}=\begin{pmatrix}1\1\1\end{pmatrix}$, $A\begin{pmatrix}1\0\1\end{pmatrix}=\begin{pmatrix}-1\0\1\end{pmatrix}$, and $A\begin{pmatrix}0\0\1\end{pmatrix}=\begin{pmatrix}1\1\2\end{pmatrix}$. If $X=(x_1,x_2,x_3)^T$ and $I$ is the identity matrix of order 3, then the system $(A-2I)X=\begin{pmatrix}4\1\1\end{pmatrix}$ has:

Accepted Answer

From the given images, the columns of $A$ can be reconstructed: the third column is $\begin{pmatrix}1\1\2\end{pmatrix}$; using $A\begin{pmatrix}1\1\0\end{pmatrix}=	ext{col}_1+	ext{col}_2=\begin{pmatrix}1\1\1\end{pmatrix}$ and $A\begin{pmatrix}1\0\1\end{pmatrix}=	ext{col}_1+	ext{col}_3=\begin{pmatrix}-1\0\1\end{pmatrix}$, one solves for columns 1 and 2.

Checking $\det(A-2I)$, one finds it is singular (zero determinant), and the given RHS vector $\begin{pmatrix}4\1\1\end{pmatrix}$ is consistent with the row space (i.e. the augmented matrix has the same rank as the coefficient matrix), so the system has infinitely many solutions.

Solutions · 1