The determinant of a `2times2` matrix
`A = ((a_1, a_2), (b_1, b_2))`
is the number `(a_1*b_2) -(a_2*b_1).` It is denoted `det A`.
In Vector Calculus, the determinant of a `2times2` matrix corresponds to the supposition of two vectors of `bbb"R"^2` in standard position `vec(a) = (a_(1), a_(2))` and `vec(b) = (b_(1), b_(2)), ` whereby we consider the matrix
`A = ((a_1, a_2), (b_1, b_2))`
that is formed by using the coordinates of each respective vector to make a row of the matrix (Bray, 25-28).
There are three options that the calculated determinant might fall under (Bray, 25-28):
This relates to the right hand rule because the right hand rule is typically used as a way to check if vectors in three-dimensions are in counterclockwise order, in which case the sign of the determinant of both a `2times2` and a `3times3` matrix is another indicator of directionality. Right-handed versus left-handed coordinates are described by Wikipedia as follows:
"For right-handed coordinates your right thumb points along the `Z` axis and the curl of your fingers represents a motion from the first or `X` axis to the second or `Y` axis. When viewed from the top or `Z` axis the system is counter-clockwise.
For left-handed coordinates your left thumb points along the `Z` axis and the curled fingers of your left hand represent a motion from the first or `X` axis to the second or `Y` axis. When viewed from the top or `Z` axis the system is clockwise."
Interestingly and sensibly, the determinant of the matrix `A` also corresponds to the area of a parallelogram formed with one vertex at the origin, one vertex at the point defined by the sum of the two vectors, and two of the edges defined by the vectors themselves. The area of this defined parallelogram is equal to the absolute value of the determinant of matrix `A`. Of course, we note that if the determinant is zero then the area of the parallelogram is also zero (Bray, 25-28).
Furthermore, the determinant of a `3times3` matrix relates to what is known as a parallelepiped, and the absolute value of the determinant is the volume of this three-dimensional shape.
Bray, Clark. Multivariable Calculus. Middletown, DE: n.p., 2015. Print.
"Right-hand Rule." Wikipedia. Wikimedia Foundation, n.d. Web. 09 June 2016.