Use row operation to find the determinant?

Question

Use row operations to find the determinant:

Can someone give me a full answer please?

Also can anyone tell me if the sign of the determinant matters ?

Row operations :

Det ( e(A) ) = -det(A) ; if e is Ri interchanged with Rj

Det ( e(A) ) = `Cdet(A) ; if e is CRi where C not equal to 0

Det( e(A) ) = det(A) ; if e is cRi + Rj

Where e i s a row operation.

Answer 1

Perform row operations to get your matrix into upper triangular form. Then the determinant is the product of the main diagonal.

Answer 2

When you perform $\mathrm R_1-\mathrm R_2$, the row $\mathrm R_1$ changes, the other two rows remain the same. $$\begin{vmatrix}3&1&2\\3&1&0\\0&1&4\end{vmatrix}$$ $\mathrm R_1-\mathrm R_2$ gives $$\begin{vmatrix}0&0&2\\3&1&0\\0&1&4\end{vmatrix}$$ When you interchange rows $\mathrm R_2$ and $\mathrm R_3$, the sign of the determinant changes. $$(-1)\begin{vmatrix}0&0&2\\0&1&4\\3&1&0\end{vmatrix}$$ For the row operations method, since you need to have an upper triangular matrix, you can exchange columns $\mathrm C_1$ and $\mathrm C_3$. In this step, just like the case of rows, the sign of the determinant would change again.

Interchanging columns $\mathrm C_1$ and $\mathrm C_2$ gives $$(+1)\begin{vmatrix}2&0&0\\4&1&0\\0&1&3\end{vmatrix}$$

And now, the product of the diagonal elements gives $6$