Determinant, Minor, Cofactor and Adjoint of a Square Matrix

1. Given a \(N \times N\) Square Matrix \(A\) having \(a_{ij}\) as elements of \(i^{th}\) Row and \(j^{th}\) Column as follows
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix}\)
   
   The Determinant of the Matrix \(A\) is given by following formula
   
   Determinant of \(A\) = \(|A|\) = \(a_{11}\) (if \(N=1\))
   
   Determinant of \(A\) = \(|A|\) = \(\sum_{j=1}^{N}(a_{1j}\hspace{.3cm} \times\)  Cofactor Value (\(a_{1j}\))) = \(\sum_{i=1}^{N}(a_{i1}\hspace{.3cm} \times\)  Cofactor Value (\(a_{i1}\)))
2. The Cofactor Value of any Element \(a_{ij}\) of the Matrix \(A\) is given by the following formula
   
   Cofactor Value (\(a_{ij}\))= \((-1)^{(i+j)}\hspace{.3cm} \times \) Minor Value (\(a_{ij}\))
3. The Minor Value of any Element \(a_{ij}\) of the Matrix \(A\) is given by the value of the Determinant of Sub-Matrix of \(A\) corresponding to the Element \(a_{ij}\).
4. The Sub-Matrix of the Matrix \(A\) corresponding to the Element \(a_{ij}\) is the Matrix obtained by Removing the Elements of \(i^{th}\) Row and \(j^{th}\) Column of Matrix \(A\).
5. A Matrix in which All the Elements of the Matrix \(A\) are Replaced by their corresponding Minor Values is known as the Minor Matrix of the Matrix \(A\).
6. A Matrix in which All the Elements of the Matrix \(A\) are Replaced by their corresponding Cofactor Values is known as the Cofactor Matrix of the Matrix \(A\).
7. The Transpose of the Cofactor Matrix of Matrix \(A\) is called the Adjoint / Adjunct / Adjugate of the Matrix \(A\).
8. Any Square Matrix whose Determinant Value is 0 is called a Singular Matrix. Any Square Matrix whose Determinant Value is Non-Zero is called a Non-Singular Matrix
9. The Sign of the Determinant Changes if any 2 Rows or Columns of the Matrix are Interchanged Odd Number of times.
10. The Sign of the Determinant Remains Unchanged if any 2 Rows or Columns of the Matrix are Interchanged Even Number of times.
11. The Absolute Value of Determinant Does Not Change on Interchanging Columns/Rows.
12. The Determinant Value of a Matrix with Odd Number of Rows and Columns Remains Unchanged if the Rows or Columns of Matrix are Rotated Cyclically.
13. The Sign of the Determinant Value of a Matrix with Even Number of Rows and Columns Changes if the Rows or Columns of Matrix are Rotated Cyclically.
14. The Determinant Value of Product of Matrices is Same as Product of the Determinant Values of each Matrix.
15. Multiplying Elements of Any 1 Row or 1 Column of the Matrix with a Constant \(k\) results in Multiplying the Determinant Value by \(k\). Multiplying All Elements of a \(N \times N\) Matrix with a Constant \(k\) results in Multiplying the Determinant Value by \(k^N\).
16. The Determinant Value of a Diagonal Matrix is equal to Product of all elements of it's Main Diagonal.
17. The Determinant Value of a \(2 \times 2\) Matrix gives the Signed Area of Parallelogram whose Adjacent Sides are given by by the 2 Vectors formed by the Rows or Columns of the Matrix in \(2\)-Dimensional Space/Plane.
18. The Determinant Value of a \(3 \times 3\) Matrix gives the Signed Volume of Parallelopiped whose Adjacent Sides are given by by the 3 Vectors formed by the Rows or Columns of the Matrix in \(3\)-Dimensional Space.
19. The Determinant Value of a \(N \times N\) Matrix gives the Signed Hyper-Volume of Parallelopiped whose Adjacent Sides are given by by the \(N\) Vectors formed by the Rows or Columns of the Matrix in \(N\)-Dimensional Space.