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If anyone can help me with the next question I would appreciate it a lot.

Let $A$ and $B$ be $n*n$ matrices and let $C=A-B$. Show that if $Ax_0=Bx_0$ and $x_0$ is not zero, then $C$ must be singular.

The first thing I don't get is the notation, what do $Ax_0$ and $Bx_0$ mean?

Thanks in advance :)

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if $x_0 \neq 0$

$$Ax_0 = B x_0$$

then we have

$$(A-B) x_0=0$$

that is we have $x_0 \neq 0$ such that $Cx_0=0$ which means $C$ is singular.

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  • $\begingroup$ I understand till $Cx_0=0$ but why is C then singular? $\endgroup$ – Amaluena Nov 28 '16 at 22:22
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    $\begingroup$ suppose $C$ is nonsingular, then I can multiply $C^{-1}$ on both sides of $Cx_0=0$ and obtain $x_0=0$ which is a contradiction. $\endgroup$ – Siong Thye Goh Nov 28 '16 at 22:29
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Presumably, $x_0$ is any non zero vector of the space where the matrices apply, and $Ax_0$, $Bx_0$ are the products of these matrices and this vector.

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If $A$ and $B$ map the vector $x_0 \ne 0$ to the same value then $$ 0 = A x_0 - B x_0 = (A-B) x_0 = C x_0 \quad (*) $$ by definition of matrix multiplication and definition of $C$.

This means $C$ has not only the zero vector as solution to $Cx=0$ but $x_0$ as well, which means that it is singular. (If there was an inverse $C^{-1}$ you would have both $0$ and $x_0$ among the choices for $C^{-1} 0$, but an inverse needs to be unique)

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  • $\begingroup$ Sorry but if $Cx_0 =0$ then why is $x_0$ a solution? $\endgroup$ – Amaluena Nov 29 '16 at 11:04
  • $\begingroup$ $x_0$ fullfils the equation $(*)$, so it is called a solution. $\endgroup$ – mvw Nov 29 '16 at 11:58
  • $\begingroup$ Oh of course! Don't know how i could miss that Thanks $\endgroup$ – Amaluena Nov 29 '16 at 14:37

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