Question

Given the following vector X, find anon zero square marix A such that AX=0;
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\[X=\begin{bmatrix}8\\6\\-7\end{bmatrix}\]
\[A=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\]

Answer 1

Definition used -
Product of two matrices -
The product of two matrices is possible if the number of columns of the first matrix is equal to the number of rows of the second matrix.
Row element of the first matrix and the corresponding column element of the second matrix multiplied and add.
Given:
\[X=\begin{bmatrix}8\\6\\-7\end{bmatrix}\]
Let \[A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}\]
Here a, b, c, d, e, f, g, and h are non-zero.
AX=0
\[\begin{bmatrix}a & b&c \\d & e&f\\g&h&i \end{bmatrix}\]
\(\displaystyle{8}{a}+{6}{b}-{7}{c}={0}\)
\(\displaystyle{8}{d}+{6}{e}-{7}{f}={0}\)
\(\displaystyle{8}{g}+{6}{h}-{7}{i}={0}\)
So we have to choose any three numbers which staifies these equations.
\(\displaystyle{a}={1},{b}={1}\) and \(\displaystyle{c}={2}\) will stsify this equation
So we can write matrix-
\[A=\begin{bmatrix}1&1&2\\-1&-1&-2\\2&2&4\end{bmatrix}\]