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Matrix is a set of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are rows and columns, the matrix is said to be an 鈥� mby n鈥� matrix, written 鈥�$m脳n$.鈥�

For,$f(x,y)=1,鈭赌(x,y)鈭�{\mathrm{鈩�}}^2,$,

We have

${鈭�}_{{惟}_2}f(x,y)dA={鈭�}_{{惟}_2}1dA=|detM|$

Which is the area of${惟}_2$.

On the other hand,

${鈭�}_{{惟}_1}g(u,v)dA={鈭�}_{{惟}_1}f\left(T\right(u,v\left)\right)dA={鈭�}_{{惟}_1}1dA=1$

Which is the area of.

From multivariable calculus, we know that, for a differentiable injective function $蠁:U鈫�{\mathrm{鈩�}}^n$where$U鈯�{\mathrm{鈩�}}^n$ is an open set, and D is the differentiation matrix of $蠁$, and a continuous function $f:蠁\left(U\right)鈫�\mathrm{鈩�}$, applies,

${鈭�}_{蠁\left(U\right)}f\left(x\right)dx={鈭�}_{U}f\left(蠁\right(u\left)\right)|detD(u\left)\right|du$.

In this case, is an invertible linear transformation, which means it's a differentiable injective function, thus we can apply the upper theorem here, giving us

$$\begin{gathered} {鈭�}_{{}_{(}{惟}_2}f(x,y)dA={鈭�}_{T_T\left({惟}_1\right)}f(x,y)dA \\ {鈭�}_{{惟}_2}f(x,y)dA={鈭�}_{{惟}_1}f\left(T\right(u,v\left)\right)|detM|dA \\ {鈭�}_{{惟}_2}f(x,y)dA=|detM|{鈭�}_{{惟}_2}g(u,v)dA \end{gathered}$$

Therefore,

${鈭�}_{{惟}_2}f(x,y)dA=|detM|{鈭�}_{{惟}_2}g(u,v)dA$.