Definitions [4]
A determinant is a single real number associated with a square matrix only.
- Denoted by det A or ∣A∣ or Δ
Consistent Solution: A system is consistent if it has at least one solution.
Inconsistent Solution: A system is inconsistent if it has no solution.
A determinant is a scalar value associated with a square matrix.
For a \[ 2 \times 2 \] matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \]
its determinant is written as: \[ |A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]
and calculated as: \[ |A| = ad - bc \]
Remember: A determinant is a number, while a matrix is an arrangement of numbers.
Consistent Solution: A system is consistent if it has at least one solution.
Inconsistent Solution: A system is inconsistent if it has no solution.
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
Cramer’s Rule
For the system: \[ a_1x + b_1y = c_1 \], \[ a_2x + b_2y = c_2 \]
\[ D = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} \]
\[ D_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} \]
\[ D_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} \]
If \[ D \neq 0, \] then the system has a unique solution:
\[ x = \frac{D_x}{D} \]
\[ y = \frac{D_y}{D} \]
Key Points
| 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 |
- A determinant is a number associated with a square matrix.
- For \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the value is \[ ad - bc. \]
- Cramer’s Rule solves a pair of linear equations using determinants.
- \[ D \] is formed from the coefficients of \[ x \] and \[ y. \]
- \[ D_x \] replaces the \[ x\text{-coefficient} \] column with constants.
- \[ D_y \] replaces the \[ y\text{-coefficient} \] column with constants.
- If \[ D \neq 0, \] the solution is \[ x = \frac{D_x}{D}, y = \frac{D_y}{D}. \]
- If \[ D = 0, \] Cramer’s Rule does not give a unique solution.
| 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 |
Concepts [7]
- Concept of Determinant
- Properties of Determinants
- Application of Determinants to Coordinate Geometry
- Methods of Solving Linear Equations in Two Variables > Determinant method (Cramer’s Rule)
- Consistency of Three Linear Equations in Two Variables
- Application of Determinants to Coordinate Geometry
- Collinearity of Three Points
