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This question is from Artin Algebra, chapter 1:

Let $A$ be a real $2\times 2$ matrix, and let $A_1$, $A_2$ be the columns of $A$. Let $P$ be the parallelogram whose vertices are $0,A_1,A_2,A_1+A_2$. Determine the effect of elementary row operations on the area of $P$, and use this to prove thatthe absolute value $|\det A|$ of the determinant of $A$ is equal to the area of $P$.

I can see what other two elementary matrices do to vector $(u,v)$, but can't see geometrically what Row-multiplying transformations do to plane. Also, although I know geometric interpretation of determinant in plane, all of them use $\sin$ or $\cos$, and I don't see how using elementary matrices, we see that determinant is related to area.

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For row multiplication, it corresponds to scaling in one of the axis direction (with reflection if the multiplier is negative). Hence the area is just being multiplied accordingly.

If a matrix is singular, then one of the columns will be multiple of the other, that is the parallelogram has degenerate shape. That is the area is $0$. Note that the determinant for a singular matrix is zero as well.

If a matrix is non-singular, note that the matrix $A$ can be written as multiplication of the elementary matrices being multiplied to the identity matrix (the reduced row echelon form). The magnitude of the determinant doesn't change under the other two elementary row operations and the area just being multiplied by the corresponding multiplier for the row multiplying operations.

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  • $\begingroup$ I am very thankful to you for this awesome reply. Although I am really sorry, that, I linked row multiplying operations, in fact, I kind of understood that, through desmos. I needed explanation for Row-addition transformations. I will be obliged if you just explain its geometric meaning for plane. Sorry, again for posting carelessly. $\endgroup$
    – Silent
    Jan 22, 2019 at 11:07
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    $\begingroup$ ah, that corresponds to shear mapping. $\endgroup$ Jan 22, 2019 at 11:34

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