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# Determine whether the following statements are true or false giving brief justifications. Any given matrix is over an arbitrary field.(a) Any system of linear equat...

## Question

###### Determine whether the following statements are true or false giving brief justifications. Any given matrix is over an arbitrary field.(a) Any system of linear equations, given by the matrix equation $A x=0$, is consistent.(b) Any system of linear equations, given by the matrix equation $A x=b$ for a fixed but arbitrary column vector $b$, is consistent.(c) Two systems of linear equations described by matrix equations, $A x=b$ and $B x=c$, are equivalent, if $B s=c$ for any solution $s$ of $A x=b$

Determine whether the following statements are true or false giving brief justifications. Any given matrix is over an arbitrary field. (a) Any system of linear equations, given by the matrix equation $A x=0$, is consistent. (b) Any system of linear equations, given by the matrix equation $A x=b$ for a fixed but arbitrary column vector $b$, is consistent.(c) Two systems of linear equations described by matrix equations, $A x=b$ and $B x=c$, are equivalent, if $B s=c$ for any solution $s$ of $A x=b$. (d) Two equivalent systems of linear equations must have the same number of equations. (e) The augmented matrix of a system of linear equations can never be a square matrix. (f) A system of linear equations consisting of a single linear equation involving $n$ variables is always consistent if $n>1$. (g) If $s_{1}$ and $s_{2}$ are solutions of the system of linear equations $A x=0$, so is $s_{1}+s_{2}$. (h) If $s_{1}$ and $s_{2}$ are solutions of the system of linear equations $A x=b$ for some non-zero column vector $b$, then so are $s_{1}+s_{2}$. (i) For a square matrix $A$, any system of linear equations $A x=b$ has a unique solution if $A$ is invertible.

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