Determinant of a matrix is a scalar value which represents the factors by which the matrix scales the Area (2D) or Volume (3D) of a region in space. Only square matrices have determinant.
This article explains the meaning of determinant, its geometrical interpretation, properties and limitations.
Determinant of a square matrix is the factor by which the matrix scales the area (2D) or volume (3D) of a region in space. It is the absolute value of the volume of the parallelepiped spanned by the column (or row) vectors of the matrix.
A positive determinant of a square matrix represents the uniform expansion of the shape (represented by the matrix) in space.
For example: If the determinant of a 2x2 matrix is 2, then the area of the shape represented by the matrix is doubled.
A negative determinant of a square matrix represents the factor by which the shape (represented by the matrix) flips in space.
For example: If the determinant of a 2x2 matrix is -2, then the area of the shape represented by the matrix flipped. The area of the space represented by -2 and 2 determinant is the same, but only the orientation is different. Its like flipping the shape.
Note: The shape only contracts if the determinant is between -1 and 1. Other wise, it expands (if determinant is greater than 1) or flips (if determinant is less than -1).
A zero determinant of a square matrix represents the redcution of the dimension of the shape (represented by the matrix) in space.
For example: If the determinant of a 2x2 matrix is 0, the area of shape is reduced to a line.
However, the zero determinant of a matrix don't tell much about dimension reduction. Matrix rank is useful to determine the dimension reduction.
There are various ways to calculate the determinant of matrices like cofactor expansion, row reduction, etc.
We will use cofactor expansion method to calculate the determinant of a matrix.
The determinant of 1 x 1 matrix is the value of the element in the matrix.
For example: If A = [5], then det(A) = 5.
The determinant of a 2x2 matrix is calculated as follows:
Let M be a 2x2 matrix:
\(M = \begin{bmatrix}
p & q \\
r & s
\end{bmatrix}
\)
\(\text{det}(M) = p \cdot s - q \cdot r
\)
The determinant of a 2x2 matrix is the product of the diagonal elements minus the product of the off-diagonal elements.
3x3 matrix is where the calculation of determinant starts to get complex.
We need to remember the method instead of the formula, because the formula is long and it gets longer as the matri size increases . But either ways we will calculate the determinant using calculator
Let M be a 3x3 matrix:
\(
M = \begin{bmatrix}
r & s & t \\
u & v & w \\
x & y & z \\
\end{bmatrix}
\)
The determinant of a 3x3 matrix is calculated as follows:
\(det(M) = r . (v . z - w . y) - s . (u . z - w . x) + t . (u . y - v . x)
\)
This method is called the cofactor expansion method. It is a recursive method.
The determinant of an NxN matrix can be calculated using the cofactor expansion method.
Let M be an NxN matrix:
\(
M = \begin{bmatrix}
m_{11} & m_{12} & m_{13} & \dots & m_{1n} \\
m_{21} & m_{22} & m_{23} & \dots & m_{2n} \\
m_{31} & m_{32} & m_{33} & \dots & m_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
m_{n1} & m_{n2} & m_{n3} & \dots & m_{nn}
\end{bmatrix}
\)
The determinant of an NxN matrix is calculated as follows:
\(\text{det}(M) = \sum_{j=1}^{n} (-1)^{1+j} \cdot m_{1j} \cdot \text{det}(M_{1j})
\)
Where M_{ij} is the matrix obtained by removing the ith row and jth column from M.
This formula is applicable for matrix of any size: 4x4, 5x5, 6x6, etc.
The complexity and time for calculation increases with the size. For large matrices, it is recommended to use a calculator to calculate the determinant
Determinant of a matrix tells a lot about matrix and how it behaves in geometrical space. Det is a very powerful concept in linear algebra and used if numerous applications.
There are some limitations of matrix determinants:
What does it mean for determinant to be between 0 and 1?
A determinant between 0 and 1 represents the contraction of the shape (represented by the matrix) in space. The area of the shape is reduced.
What does it mean for determinant to be 1?
A determinant of 1 represents the shape is not expanded or contracted. The area of the shape remains the same.
What does it mean for determinant to be -1?
A determinant of -1 represents the shape is flipped. The area of the shape remains the same, but the orientation is flipped.