# Invertibility of Matrix by examining the column vectors

Here is my question:

Give a matrix B, determine the invertibility of B by examining the column vectors of B.

I have solved this one way, by trying to show that the column vectors are independent, by computing the determinant of the matrix B, if it is NON-Zero then for this situation of being a homogeneous system, then we get that we have the trivial solution, which means the coefficients of the system of vectors based on the columns of the matrix B are all zero, which means the vectors are independent.

I dont know if this question allows you to compute the determinant. If we are not allowed, how else can one show that the columns are independent. Do we need to perform row-echelon form reduction on the matrix. Also the very specific matrix that was given, the first element of this 3x3 has a zero.

Hope someone can tell me how to go about showing this.

Paul

• If the first element of the matrix is zero you can permute two lines. – mfl Jun 24 '14 at 18:53
• Hi mfl, I actually permuted the rows so i get something in close to echelon form, then I apply row-reduction to this matrix. And i guess since this linear system is supposed to equal zero,because it forms a homogeneous system, I was able to reduce it to the identity matrix. – Palu Jun 24 '14 at 19:13
• BUT, when I computed the determinant of the original matrix, i got a value of 9. But when I did rref, this final matrix has a determinant of 1. I thought that both original and rref matrix, their determinants would always be the same, why is it not the same? – Palu Jun 24 '14 at 19:15
• It is the same if you add to a row a multiple of other row, that is, $R_2-3R_1\rightarrow R_2.$ But if you multiply some row by a number and add to this row (a multiple of) another row then you are changing the determinant. That is, if $2R_2-R_1\rightarrow R_2$ then the determinant is multiplied by $2.$ – mfl Jun 24 '14 at 19:33
• OK, i forgot about that about determinant properties. So as long as I can apply gauss-jordan and achieve the Identity then that proves that the columns are Independent? – Palu Jun 24 '14 at 20:09