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Let A be a nxn matrix, x be a nx1 matrix and 0 be a nx1 zero matrix.
Ax=0

a)If A is invertible, solve the homogeneous system of linear equations

b)Is Ax = 0 always consistent? Explain

Can anyone start me off how to go about to solving this? I am not too sure about a)

For b), I wrote Yes as a homogeneous equation is always consistent.There is always a solution of x=0, hence it will always be consistent.Am I right?

c00kie132
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    For part a, start with the definition of a matrix inverse. Think how you can use that definition, and you can always do the same thing to both sides of an equation. – Golden_Ratio Jan 22 '22 at 08:53
  • @Golden_Ratio so mutiplying A^-1 to both sides , x will still be equal to 0. Thats the answer? – c00kie132 Jan 22 '22 at 10:21

1 Answers1

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a) If A is invertible, it means that $Ax = 0$, for $x$ only has $0$ for the answer.

You can use gaussian elimination or inverse matrix to solve it.

$Ax = 0$

$A^{-1}Ax = A^{-1}×0$

$x = 0$

b) Of course $Ax = 0$ always has the answer, and that is $0$.

Reference

$A$ is invertible matrix iff $Ax=0$ has the trivial solution only.

jackson
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