Overview of Determinants

A determinant is a single real number associated with a square matrix only.

• Denoted by det ⁡A or ∣A∣ or Δ 

Determinant of Order 1:

If A = [a] then det⁡A = ∣a∣ = a

Determinant of Order 2:

\[\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}\] = ad − bc​

Determinant of Order 3:

\[\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{vmatrix}\]

\[a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1(a_2b_3-a_3b_2)\]

Applicable ONLY for 3×3 determinants

Steps:

1. Rewrite the first two columns to the right

2. Add products of downward diagonals

3. Subtract products of upward diagonals

Minors:

Minor of element a_ij is the determinant obtained after deleting the i-th row and j-th column.

Cofactors:

If  M_ij is the minor of the element a_ij in the determinant Δ, then the number (−1)^i+j​ M_ij is called the cofactor of the element a_ij; it is usually denoted by A_ij.

Thus A_ij = (−1)^i+j M_ij​

Corollaries:

• Passing a row (or column) over nnn rows (or columns):

  Δ₁ = (−1)ⁿΔ

• If each element of a determinant of order nnn is multiplied by k:

  Δ₁ = kⁿΔ

• If two rows or columns are proportional:

  Δ = 0

• If A is skew-symmetric of odd order:

  ∣A∣ = 0

• Polynomial result: If Δ = 0 when x = a, then

  (x − a) is a factor

1. Area of a Triangle:

\[\mathrm{Area}=\frac{1}{2}
\begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{vmatrix}\]

2. Collinearity of Three Points:

\[\begin{vmatrix}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{vmatrix}=0\]

1. Adjoint of a Square Matrix

The adjoint of a square matrix A is the transpose of the matrix of cofactors of A.

2. Fundamental Identity

A(adj⁡A) = ∣A∣I = (adj⁡A)A

3. Inverse of a Matrix
If ∣A∣ ≠ 0, then:

\[A^{-1} =\frac{1}{|A|}\operatorname{adj}A\]

4. Singular and Non-Singular Matrix

• Singular matrix: ∣A∣ = 0

• Non-singular matrix: ∣A∣ ≠ 0

5. Invertibility Condition

A is invertible   ⟺  ∣A∣ ≠ 0

System of linear equations: AX = B

Consistent / Inconsistent:

• Consistent → one or more solutions

• Inconsistent → no solution

Matrix method (Martin’s Rule):

If ∣A∣ ≠ 0, X = A⁻¹B ⇒ unique solution

When ∣A∣ = 0:

• (adjA)B = 0 → infinitely many solutions

• (adj⁡A)B ≠ 0 → no solution

Homogeneous system:

AX = 0

• Always consistent

• ∣A∣ ≠ 0 → trivial solution

• ∣A∣ = 0→ infinitely many solutions