- 362

- 0

Theorem 2.3.3

A homogeneous system of

So in other words, if no row of the matrix consists entirely of zeros, then the only solution is X=0.

This is what my book teaches me before determinants. When it introduces determinants, it asks as an exerice to rewrite the theorem using the determinant.

The book rewrites it,

...iff the determinant does not equal zero.

p->q

p->r

q->p

r->p

Thus

r->q and q->r

Which means that if there are no rows consisting entirely of zeros, then the determinant is not zero. This is clearly false, since a determinant can be zero if two rows are proportional. I feel like my book has lied to me again.

I at first trusted my book over my own belief, and when it told me to determine for what values of a (some coefficient) does this system has nontrivial solutions, I set up an upper-triangular determinant, multiplied the elements along the principal diagonal and set it equal to zero. This process yields solutions which are trivial. Admittedly, it would have been a lot easier to set the last element in the last row equal to zero, but I thought this would omit some possibilities.

For instance,

x+y=ax

-x+y=ay

Using the above procedure, I get a=1 as one of the possibilities.

Am I completely off with this accusation?

A homogeneous system of

*m*linear algebraic equations in*m*unknowns has no nontrivial solutions iff the reduced coefficient matrix has no rows consisting entirely of zeros.So in other words, if no row of the matrix consists entirely of zeros, then the only solution is X=0.

This is what my book teaches me before determinants. When it introduces determinants, it asks as an exerice to rewrite the theorem using the determinant.

The book rewrites it,

...iff the determinant does not equal zero.

p->q

p->r

q->p

r->p

Thus

r->q and q->r

Which means that if there are no rows consisting entirely of zeros, then the determinant is not zero. This is clearly false, since a determinant can be zero if two rows are proportional. I feel like my book has lied to me again.

I at first trusted my book over my own belief, and when it told me to determine for what values of a (some coefficient) does this system has nontrivial solutions, I set up an upper-triangular determinant, multiplied the elements along the principal diagonal and set it equal to zero. This process yields solutions which are trivial. Admittedly, it would have been a lot easier to set the last element in the last row equal to zero, but I thought this would omit some possibilities.

For instance,

x+y=ax

-x+y=ay

Using the above procedure, I get a=1 as one of the possibilities.

Am I completely off with this accusation?

Last edited: