Can we use $R_1 \to R_2-R_1$ as an elementary row operation without changing the value of the determinant?

Question

Can someone help me out? I need to know if $R_1 \to R_2-R_1$ is a valid elementary row operation that can be used on the given determinant without changing the determinant's value. It's a $3\times 3$ determinant. Whenever I do this, I'm getting a negative value of the determinant.

Here $R_1 \to R_2 - R_1$ is replacing the elements of the row $1$ by subtracting the elements row $1$ from the corresponding elements of row $2$.

Answer 1

No, the elementary row operation would be $R1\to R1-R2$.

Answer 2

There are three types of elementary row operation:

• Row swap: $R_i \to R_j$, $R_j \to R_i$
• Row (scalar) multiplication: $R_i \to kR_i$, $k \in \mathbb{R}$, $k\neq 0$
• Row addition: $R_i \to R_i + kR_j$, $k\in \mathbb{R}$, $k \neq 0$

Your proposed operation doesn't appear in the above list, so the operation $R_1 \to R_2 - R_1$ is not an elementary row operation. However, it is the composition of two elementary row operations: $R_1 \to R_1 - R_2$ followed by $R_1 \to - R_1$.

The effect that the operation $R_1 \to R_2 - R_1$ has on the determinant is the combined effects of the elementary row operations $R_1 \to R_1 - R_2$ and $R_1 \to - R_1$. Recall that:

• a row swap multiplies the determinant by $(-1)$,
• a row multiplication by $k$ multiplies the determinant by $k$, and
• a row addition has no effect on the determinant.

As $R_1 \to R_1 - R_2$ has no effect on the determinant, and $R_1 \to -R_1$ multiplies the determinant by $-1$, the overall effect of the operation $R_1 \to R_2 - R_1$ on the determinant of the matrix is multiplication by $-1$. In particular, unless the matrix has determinant zero, the operation $R_1 \to R_2 - R_1$ does change the value of the determinant.