Answer A,C, D
$Ax = \lambda x$
Learn very powerful and important way to multiply matrix and vector. $Ax$ is just linear combination of columns of $A$.. if you can understand this then solution of system of linear equation is just cakewalk for you. And complete linear algebra will come very intutive.
A.
$ \Bigg[\begin{smallmatrix}
-9 & -6 & -2 & -4
\\ -8 & -6 & -3 & -1
\\ 20 & 15 & 8 & 5
\\ 32 & 21 & 7 & 12
\end{smallmatrix} \Bigg]
\Bigg[\begin{smallmatrix}
-1\\ 1\\ 0 \\1
\end{smallmatrix} \Bigg] =
-1 \Bigg[\begin{smallmatrix}
-9\\ -8\\ 20 \\32
\end{smallmatrix} \Bigg] + 1
\Bigg[\begin{smallmatrix}
-6\\ -6\\ 15 \\21
\end{smallmatrix} \Bigg]
+ 0 \Bigg[\begin{smallmatrix}
-2\\ -3\\ 8 \\7 \end{smallmatrix} \Bigg]
+ 1 \Bigg[\begin{smallmatrix}
-4\\ -1\\ 5 \\12
\end{smallmatrix} \Bigg] =
\Bigg[\begin{smallmatrix} -1\\1 \\ 0 \\1 \end{smallmatrix} \Bigg]$
B.
$ \Bigg[\begin{smallmatrix}
-9 & -6 & -2 & -4
\\ -8 & -6 & -3 & -1
\\ 20 & 15 & 8 & 5
\\ 32 & 21 & 7 & 12
\end{smallmatrix} \Bigg]
\Bigg[\begin{smallmatrix}
1\\ 0\\ -1 \\0
\end{smallmatrix} \Bigg] =
1 \Bigg[\begin{smallmatrix}
-9\\ -8\\ 20 \\32
\end{smallmatrix} \Bigg] + 0
\Bigg[\begin{smallmatrix}
-6\\ -6\\ 15 \\21
\end{smallmatrix} \Bigg]
-1 \Bigg[\begin{smallmatrix}
-2\\ -3\\ 8 \\7 \end{smallmatrix} \Bigg]
+ 0 \Bigg[\begin{smallmatrix}
-4\\ -1\\ 5 \\12
\end{smallmatrix} \Bigg] =
\Bigg[\begin{smallmatrix} -7\\-5 \\ 12 \\25 \end{smallmatrix} \Bigg]$
C.
$ \Bigg[\begin{smallmatrix}
-9 & -6 & -2 & -4
\\ -8 & -6 & -3 & -1
\\ 20 & 15 & 8 & 5
\\ 32 & 21 & 7 & 12
\end{smallmatrix} \Bigg]
\Bigg[\begin{smallmatrix}
-1\\ 0\\ 2 \\2
\end{smallmatrix} \Bigg] =
-1 \Bigg[\begin{smallmatrix}
-9\\ -8\\ 20 \\32
\end{smallmatrix} \Bigg] + 0
\Bigg[\begin{smallmatrix}
-6\\ -6\\ 15 \\21
\end{smallmatrix} \Bigg]
+ 2 \Bigg[\begin{smallmatrix}
-2\\ -3\\ 8 \\7 \end{smallmatrix} \Bigg]
+ 2 \Bigg[\begin{smallmatrix}
-4\\ -1\\ 5 \\12
\end{smallmatrix} \Bigg] =
3\Bigg[\begin{smallmatrix} -1\\ 0\\ 2\\2 \end{smallmatrix} \Bigg]
$
D.
$ \Bigg[\begin{smallmatrix}
-9 & -6 & -2 & -4
\\ -8 & -6 & -3 & -1
\\ 20 & 15 & 8 & 5
\\ 32 & 21 & 7 & 12
\end{smallmatrix} \Bigg]
\Bigg[\begin{smallmatrix}
0\\ 1\\ -3 \\0
\end{smallmatrix} \Bigg] =
0 \Bigg[\begin{smallmatrix}
-9\\ -8\\ 20 \\32
\end{smallmatrix} \Bigg] + 1
\Bigg[\begin{smallmatrix}
-6\\ -6\\ 15 \\21
\end{smallmatrix} \Bigg]
-3 \Bigg[\begin{smallmatrix}
-2\\ -3\\ 8 \\7 \end{smallmatrix} \Bigg]
+ 0 \Bigg[\begin{smallmatrix}
-4\\ -1\\ 5 \\12
\end{smallmatrix} \Bigg]
= 3\Bigg[\begin{smallmatrix} 0\\ 1\\ -3 \\0 \end{smallmatrix} \Bigg]$
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