The determinant

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• Linear Transformations tend to increase or decrease the area of a given space.
• The Determinant is actually the area between 2 given vectors, since the area between the basis vectors is 1, and a matrix transformation is just where the basis vectors land, the determinant of a matrix provides the factor by which the area between any 2 vectors is scaled.
• If a transformation smushes a 2D plane into a single line, then its determinant is 0
• if a determinant of a transformation is 0, it scales down the space into a lower dimension.
• Invert the orientation of space when the direction of the space changes, usually $\hat{i}$ is to the right of $\hat{j}$, but if a transformation changes this and $\hat{i}$ is on left of $\hat{j}$ that is …
• The determinant of a transformation, which inverts the orientation of space will be negative
• Why negative number? the determinant is positive if $\hat{i}$ is to the right of $\hat{j}$, and as it approach closer to $\hat{j}$ the determinant reduces, and reaches 0 when both $\hat{i}\text{ and }\hat{j}$ aligns. So when $\hat{i}$ moves past $\hat{j}$ it is denoted by negative number

In 2 Dimension the determinant is a measure of area, and in 3 Dimension it is a measure of volume.

How to compute the determinant:

$$\det (\left[\begin{array}{c}ab\\ cd\end{array}\right])=ad-bc$$

Why do we subtract them? because,a gives the co-ordinate of $\hat{i}$ in the X axis and d gives the co-ordinate of $\hat{j}$ in the Y axis, since both of them are in their corresponding axis, they increase the size of the area, but when the $\hat{i}$ or $\hat{j}$ moves out of their respective X or Y axis, the area reduces, that is when their numbers are positive (Think of our 2D plane and the number lines on the each axis, this would make sense)

det($M_1{M}_2$) = det($M_1$) x det($M_2$)