Definitions [6]
A determinant is a single real number associated with a square matrix only.
- Denoted by det A or ∣A∣ or Δ
Let A = [aij] be a square matrix of order n. Then, the minor Mij of aij in A is the determinant obtained by deleting the ith row and the jth column in which element aij lies. It is denoted by Mij of A.
Let A = [aij] be a square matrix of order n. Then, the cofactor Cij (or Aij) of aij in A is (−1)i+j times Mij, where Mij is the minor of aij in A.
∴ Cij = (−1)i+j Mij
Consistent Solution: A system is consistent if it has at least one solution.
Inconsistent Solution: A system is inconsistent if it has no solution.
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).
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.
Formulae [2]
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
For \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \]
\[ \boxed{\text{adj } A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}} \]
Shortcut
For a \[2 \times 2\] matrix:
- Interchange the diagonal elements.
- Change the signs of the off-diagonal elements.
Key Points
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Minor \[M_{ij}\]: determinant of the matrix obtained by deleting row i and column j.
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Cofactor \[C_{ij}\]: \[C_{ij} = (-1)^{i+j}M_{ij}\].
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Determinant expansion along row i: \[|A| = \sum_{j=1}^{n} a_{ij}C_{ij}\].
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Determinant expansion along column j: \[|A| = \sum_{i=1}^{n} a_{ij}C_{ij}\].
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Determinant value is the same for any choice of row or column for expansion.
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Mixed row/column property: \[\sum_{j=1}^{n} a_{ij}C_{kj} = 0\] for \[i \neq k\].
| Concept | Key Point / Formula |
|---|---|
| Area of Triangle | \[ \boxed{\dfrac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}} \] |
| Collinearity | Three points are collinear if determinant =0=0 |
| Equation of Line | Line through two points can be written using a 3 × 3 determinant |
| Consistent System | Has at least one solution |
| Inconsistent System | Has no solution |
| Non-Singular Case | \[ |A| \neq 0 \] |
| Singular Case | ∣A∣=0 |
| Concept | Formula / Rule |
|---|---|
| Adjoint | adjA= transpose of cofactor matrix |
| 2×2 Adjoint | \[ \text{adj} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \] |
| Fundamental Property | \[ A(\text{adj } A) = (\text{adj } A)A = |A|I_n \] |
| Singular Matrix | if ∣A∣ = 0 |
| Non-Singular Matrix | if \[ |A| \neq 0 \] |
| For a square matrix A of order n | \[ |\text{adj } A| = |A|^{n-1} \] |
| Invertibility | A is invertible iff \[ |A| \neq 0 \] |
| Inverse | \[ A^{-1} = \frac{1}{|A|}\ \text{adj } A \] |
Concepts [13]
- Concept of Determinant
- Minors and Co-factors
- Theorems on Determinants
- Properties of Determinants
- Application of Determinants to Coordinate Geometry
- Simplification of Determinants using Properties
- Adjoint and Inverse of a Matrix
- Inverse of a Square Matrix
- Singular and Non-singular Matrix
- Relations between a Matrix and its Adjoint
- Inverse of a Square Matrix
- Properties of Invertible Matrices
- Solution of System of Linear Equations
