Revision: Algebra >> Determinants Maths Commerce (English Medium) Class 12 CBSE

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

• Denoted by det ⁡A or ∣A∣ or Δ 

To find the determinant, multiply each element of your chosen row (or column) by its corresponding sign multiplier \[(-1)^{i+j}\] and the \[2 \times 2\] determinant that remains after deleting that element's row and column.

Consistent Solution: A system is consistent if it has at least one solution.

Inconsistent Solution: A system is inconsistent if it has no solution.

Let A = [a_ij] be a square matrix of order n. Then, the cofactor C_ij (or A_ij) of a_ij in A is (−1)^i+j times M_ij, where M_ij is the minor of a_ij in A.

∴ C_ij = (−1)^i+j M_ij

Let A = [a_ij] be a square matrix of order n. Then, the minor M_ij of a_ij in A is the determinant obtained by deleting the i^th row and the j^th column in which element a_ij lies. It is denoted by M_ij​ of A.

If A and B are non-singular square matrices of the same order such that AB = BA = I (where I is the identity matrix of the same order as A and B), then A and B are called inverses of each other.

We write A⁻¹ = B and B⁻¹ = A.

i.e. AA⁻¹ = A⁻¹A = I.

• If |A| ≠ 0, then A⁻¹ exists.
• If the inverse of a square matrix exists, then it is unique. A matrix can not have more than one distinct inverse.

The adjoint of A is defined as the transpose (i.e. interchange rows and columns) of the cofactor matrix, and it is denoted by adj (A).

Consistent Solution: A system is consistent if it has at least one solution.

Inconsistent Solution: A system is inconsistent if it has no solution.

Order 1 (1×1 matrix):

∣A∣ = a

Order 2 (2×2 matrix):

∣A∣ = ad − bc

Order 3 (3×3 matrix):

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

\[|A|=a_{11}(a_{22}a_{33}-a_{32}a_{23})-a_{12}(a_{21}a_{33}-a_{31}a_{23})+a_{13}(a_{21}a_{32}-a_{31}a_{22})\]

• If |A| = 0
  A matrix is called a Singular Matrix
• If |A| ≠ 0
  Matrix is called a Non-Singular Matrix

By Adjoint Method: \[A^{-1}=\frac{\mathrm{adj}A}{|A|}\]

By Using Algebraic Equation: A matrix A and an algebraic equation in matrix A is in the form of A² + bA + C = O.

\[A^{-1}=\frac{1}{C}\left[-aA-bI\right]\]

• Area of triangle using determinant:

• Expanded coordinate form:

  \[\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\].

• Area is always taken as positive; use absolute value.

• For collinear points, determinant = 0, so area = 0.

• Minor \[M_{ij}\]: determinant of the matrix obtained by deleting row i and column j.

• Cofactor \[C_{ij}\]: \[C_{ij} = (-1)^{i+j}M_{ij}\].

• Determinant expansion along row i: \[|A| = \sum_{j=1}^{n} a_{ij}C_{ij}\].

• Determinant expansion along column j: \[|A| = \sum_{i=1}^{n} a_{ij}C_{ij}\].

• Determinant value is the same for any choice of row or column for expansion.

• Mixed row/column property: \[\sum_{j=1}^{n} a_{ij}C_{kj} = 0\] for \[i \neq k\].

1. adj (AB) = (adj B) (adj A)
2. (adj A)A = A (adj A) = |A| Iₙ
3. (a) |adj A| = |A|ⁿ⁻¹, if |A| ≠ 0
   (b) |adj A| = 0, if |A| = 0
4. If |A| = 0, then (adj A) A = A (adj A) = O
5. adj (Aᵐ) = (adj A)ᵐ, m ∈ N
6. adj (kA) = kⁿ⁻¹ (adj A), k ∈ R
7. adj (adj A) = |A|ⁿ⁻² A, A is non-singular matrix
8. adj (adj A) = |A|ⁿ⁻² A, A is non-singular matrix

• Area of triangle using determinant:

• Expanded coordinate form:

  \[\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|\].

• Area is always taken as positive; use absolute value.

• For collinear points, determinant = 0, so area = 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

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