Determinant of a Matrix (Geometrical Interpretation)

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.

Geometrical Interpretation of Determinant

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.

Positive Determinant (Expansion)

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.

Negative Determinant (Shape Flipped)

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).

Zero Determinant (Dimension Collapse)

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.

How to Calculate Determinant of a Matrix

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.

1 x 1 Matrix Determinant

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.

2x2 Matrix Determinant

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 Determinant

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.

NxN Matrix Determinant

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

Properties of Determinant

• Determinant is a scalar value. It is a single number.
• Determinant of a matrix is unique, its always same irrespective the method used to calculate it.
• Determinant exists only for square matrix which is also non singular matrix.
• Determinant of a singular matrix is zero.
• Determinant of a matrix is the product of the eigen values of the matrix.
• Determinant of a matrix is the area (2D) or volume (3D) scaling factor of the matrix.
• Determinant of a matrix is the factor by which the matrix scales the area or volume of a region in space.
• The value of determinant tells either the shape (represented by matrix in plane) expands (det > 1), contracts (0 < det < 1) or flips (det < -1) in space.
• Determinant of matrix is equal to the determinant of its transpose.
  

Limitations of 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:

• Determinant of a matrix don't exist for non-square matrices, as non-square matrix represent transformation between different dimensions.
• Determinant of a matrix is not defined for singular matrices, complex matrices and infinite matrices.
• Determinant of a matrix being 0, don't tell much about the matrix. It only tells the matrix is singular. Matrix rank is useful to determine the dimension reduction.

Frequently Asked Questions

1. 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.

2. 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.

3. 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.