Effect of row augmentation on value of determinant.

Question

Part(a) is done. How to proceed for part (b).

My first question is what do they mean by row augmentation ?

Do they mean the row operation of adding k times the first row to third by row augmentation ?

Answer 1

I have solved my question.

Thanks everyone.

Answer 2

Here are three relevant rules determinant manipulation:

Let $A$ be a square matrix. Let $B$ be made from $A$ by

$(1)$ Multiplication of a row of $A$ with a scalar $c$; then $\det B = c\det A$.
$(2)$ Swapping two rows; then $\det B = -\det A$.
$(3)$ Adding a multiplum of a row to another row; then $\det B = \det A$.

Which one is relevant for you?

Answer 3

An idea: The row augmentation on $\;A\;$ is then the same as the product $\;GA\;$. Now, why not using the all important theorem that for any two square matrices $\;X,Y\;$ of the same order we have that $\;\det(XY)=\det X\cdot\det Y\;$ ?