Determinants & Properties

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Introduction

The determinant is a scalar value associated with a square matrix.
It tells us important information such as:

  • Whether a matrix is invertible
  • Volume scaling of linear transformations
  • Whether a system of linear equations has a unique solution
  • Orientation (sign) of transformation

A determinant is defined only for square matrices.

Determinant of a 2 × 2 Matrix

For a matrix

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

its determinant is

\det(A) = ad - bc

This is the simplest and most commonly used determinant formula.

Determinant of a 3 × 3 Matrix

Given

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}

The determinant is:

\det(A) = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}

Each 2 × 2 block is a minor, and the sign pattern follows cofactor expansion.

General Determinant Expansion (Cofactor Expansion)

For the i-th row:

\det(A) = \sum_{j=1}^{n} a_{ij} , C_{ij}

where C_{ij} is the cofactor:

C_{ij} = (-1)^{i+j} M_{ij}

and M_{ij} is the determinant of the submatrix obtained by deleting row i and column j.

Elementary Properties of Determinants

1. Determinant changes sign when two rows are interchanged

If you swap any two rows:

\det(A_{\text{swapped}}) = -\det(A)

2. If two rows (or columns) are identical → determinant is zero

This means the matrix is singular.

3. If a row or column is all zeros → determinant is zero

4. Multiplying a row by a scalar multiplies determinant by that scalar

If a row is multiplied by k:

\det(A_{\text{new}}) = k \cdot \det(A)

5. Adding a multiple of one row to another row does not change the determinant

This property is used heavily in row-reduction.

6. Determinant of a triangular matrix = product of its diagonal entries

For upper/lower triangular matrices:

\det(A) = a_{11} , a_{22} , a_{33} , \dots , a_{nn}

This is also used to quickly compute determinants during row operations.

7. Determinant of the identity matrix is 1

\det(I) = 1

8. Determinant of a matrix product

\det(AB) = \det(A)\det(B)

This property is important in eigenvalue theory.

9. Determinant of a transpose

\det(A^T) = \det(A)

Geometric Interpretation

1. Area in 2D

For vectors \vec{u} = (a, b) and \vec{v} = (c, d):

\text{Area} = \begin{vmatrix} a & b \\ c & d \end{vmatrix}

Absolute value gives the area of the parallelogram formed.

2. Volume in 3D

For vectors \vec{a}, \vec{b}, \vec{c}:

\text{Volume} = \begin{vmatrix} a_x & a_y & a_z \\ b_x & b_y & b_z \\ c_x & c_y & c_z \end{vmatrix}

This equals the scalar triple product.

When is a Matrix Invertible?

A matrix is invertible if and only if its determinant is non-zero.

\det(A) \neq 0 \quad \Rightarrow \quad A^{-1} \text{ exists}

If \det(A) = 0, the matrix:

  • is singular
  • does not have an inverse
  • represents linearly dependent rows/columns
  • leads to no unique solution in a system of equations

Determinant and System of Linear Equations (Cramer’s Rule)

For the system AX = B, if \det(A) \neq 0, the solution is:

x_i = \frac{\det(A_i)}{\det(A)}

where A_i is obtained by replacing the i-th column of A with B.

Conclusion

Determinants are more than just computational tools — they reveal deep information about a matrix: invertibility, linear dependence, volume scaling, and geometric transformation behavior. Understanding determinant properties will make future topics like rank, eigenvalues, and linear systems much easier.

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