Suppose that A is an nxn matrix with columns C1; C2; C3... Cn. If:
2C1 + 3C2 - C3 = 0
show that A is not invertible.

Question

Suppose that A is an nxn matrix with columns C1; C2; C3... Cn. If:
2C1 + 3C2 - C3 = 0
show that A is not invertible.

helder_edwin · Answer

are u allowed to use determinants?

anonymous · Answer

Yeah

helder_edwin · Answer

well then it is easy. the equation
$$ \large 2C_1+3C_2-C_3=0 $$
means that (at least) three columns are linearly dependant so the determinant is zero so the matrix is not invertible.

anonymous · Answer

Thanks

helder_edwin · Answer

u r welcome

anonymous · Answer

How would it be done without using determinants?

helder_edwin · Answer

the rank of a matrix is the dimension of both the row and column spaces. for a $n\times n$ matrix to be invertible its rank has to be =n. since
$$ \large 2C_1+3C_2-C_3=0 $$
then the rank of this matrix is less than n. so the matrix cannot be invertible.