C4 Maths Part 2

i) Minors and Cofactors of a matrix ii) Properties of determinants iii) Determinants of different types of matrices

12th

1. Fundamental Definitions

• Determinant: A number (real or complex) associated with every square matrix $A$. It is denoted by $|A|$, $\det(A)$, or $\Delta$.

• Order 1: If $A = [a]$, then $|A| = a$.

• Order 2: If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $|A| = ad - bc$.

• Order 3: Calculated by expanding along any row or column. Usually, the row/column with the most zeros is chosen to simplify the calculation.

2. Minors and Cofactors

• Minor ($M_{ij}$): The determinant obtained by deleting the $i^{th}$ row and $j^{th}$ column of an element $a_{ij}$.

• Cofactor ($A_{ij}$): Defined as $A_{ij} = (-1)^{i+j} M_{ij}$.

• Key Property: The sum of products of elements of any row (or column) with their corresponding cofactors equals the determinant value. If multiplied by cofactors of a different row/column, the sum is zero.

3. Properties of Determinants

• Transposition: $|A'| = |A|$, meaning the value remains unchanged if rows and columns are interchanged.

• Interchange: If any two rows (or columns) are interchanged, the sign of the determinant changes.

• Identical Rows: If any two rows or columns are identical or proportional, the determinant is zero.

• Scalar Multiplication: $|kA| = k^n |A|$, where $n$ is the order of the matrix.

• Product Rule: $|AB| = |A||B|$ for square matrices $A$ and $B$ of the same order.

4. Adjoint and Inverse

• Adjoint ($\text{adj } A$): The transpose of the matrix formed by the cofactors of the elements of $A$.

  • $A(\text{adj } A) = (\text{adj } A)A = |A|I$.

• Singular Matrix: A matrix where $|A| = 0$.

• Non-Singular Matrix: A matrix where $|A| \neq 0$. Only non-singular matrices are invertible.

• Inverse Formula: $A^{-1} = \frac{1}{|A|} (\text{adj } A)$.

5. Applications

• Area of a Triangle: For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, the area is given by:

  $\text{Area} = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$

  If the area is zero, the points are collinear.

• Solving Linear Equations (Matrix Method): For a system $AX = B$

  • If $|A| \neq 0$, there is a unique solution: $X = A^{-1}B$ (Consistent).

  • If $|A| = 0$ and $(\text{adj } A)B = 0$, the system has infinitely many solutions.

  • If $|A| = 0$ and $(\text{adj } A)B \neq 0$, the system has no solution (Inconsistent).

Exercise analysis