Summary

Key Concepts and Formulas

1. Determinant Expansion

• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $|A| = ad - bc$.
• For a $3 \times 3$ matrix, the determinant is expanded using the elements of any row or column and their corresponding cofactors. E.g., expanding along $R_1$: $\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$.

2. Area of a Triangle

• Area $\Delta = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right|$.
• Collinearity: Three points are collinear if the area of the triangle formed by them is zero, i.e., $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$.
• Equation of a line: The equation of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0$.

3. Minors and Cofactors

• Minor ($M_{ij}$): Determinant obtained by deleting the $i^{th}$ row and $j^{th}$ column.
• Cofactor ($A_{ij}$): $A_{ij} = (-1)^{i+j} M_{ij}$.
• Zero Property: The sum of products of elements of any row (or column) with the cofactors of any other row (or column) is zero.

4. Adjoint and Inverse of a Matrix

• Adjoint ($\text{adj } A$): The transpose of the cofactor matrix.
• Inverse ($A^{-1}$): $A^{-1} = \frac{1}{|A|} \text{adj } A$, provided $|A| \neq 0$ (i.e., $A$ is non-singular).

Important Adjoint Properties (Order $n$)

1. $A(\text{adj } A) = (\text{adj } A)A = |A|I$
2. $|\text{adj } A| = |A|^{n-1}$
3. $|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$
4. $\text{adj}(AB) = (\text{adj } B)(\text{adj } A)$

Important Inverse Properties

1. $(A^{-1})^{-1} = A$
2. $(AB)^{-1} = B^{-1}A^{-1}$ (Reversal Law)
3. $(A^T)^{-1} = (A^{-1})^T$
4. $|A^{-1}| = \frac{1}{|A|}$

5. System of Linear Equations ($AX = B$)

• Unique Solution (Consistent): If $|A| \neq 0$, then $X = A^{-1}B$.
• No Solution (Inconsistent): If $|A| = 0$ and $(\text{adj } A)B \neq O$.
• Infinitely Many Solutions (Consistent): If $|A| = 0$ and $(\text{adj } A)B = O$.

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Exam Tips for Board Exams

Show Your Work

Board exams award step marks. When finding the inverse of a $3 \times 3$ matrix, explicitly show the calculation of at least a few cofactors. Do not just write the final adjoint matrix directly.

Formula First

Always write the general formula before plugging in numbers (e.g., write $A^{-1} = \frac{1}{|A|} \text{adj } A$ before calculating it).

Checking Consistency

Before calculating the adjoint for solving $AX = B$, always calculate $|A|$ first. If $|A| = 0$, explicitly state the condition you are checking next ($(\text{adj } A)B$) and write the conclusion clearly.

Matrix Polynomials

For questions asking to prove $A^2 - 5A + 7I = O$ and "hence find $A^{-1}$", do not use the standard adjoint method to find $A^{-1}$. You must multiply the polynomial equation by $A^{-1}$ and solve it algebraically. Using the adjoint method here will result in a loss of marks.

Word Problems

Clearly define your variables ($x, y, z$) and formulate the equations step-by-step. Representing them in the $AX=B$ format carries partial credit even if subsequent calculations have an error.

Smart Expansion

When evaluating a $3 \times 3$ determinant, scan for the row or column with the maximum number of zeros and expand along it to save time and reduce calculation errors.

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Exam Tips for JEE Main & Advanced

Master Adjoint Properties

The properties $|\text{adj } A| = |A|^{n-1}$ and $|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$ are extremely high-yield. Memorize them perfectly. They often turn complex 3-minute calculations into 10-second mental math problems.

Homogeneous Systems ($AX = O$)

Questions frequently ask for conditions where a homogeneous system has non-trivial solutions. This directly implies you need to set $|A| = 0$ and solve for the unknown parameter (like $\lambda$ or $k$).

Trigonometric & Polynomial Determinants

JEE questions often embed trigonometry or polynomials inside determinants. Use row/column operations ($R_i \to R_i + kR_j$) vigorously to create zeros or pull out common factors before expanding. Expanding directly usually leads to messy algebra.

Cayley-Hamilton Theorem

While not explicitly in the NCERT syllabus, knowing that every square matrix satisfies its own characteristic equation ($|A - \lambda I| = 0$) is a massive shortcut for finding higher powers of a matrix (like $A^{50}$) or relating $A^{-1}$ to $A$ and $A^2$.

Symmetric and Skew-Symmetric Integration

Be prepared to use determinant properties like $|A^T| = |A|$. Recall that the determinant of an odd-order skew-symmetric matrix is always $0$.

Check for Constant Determinants

Sometimes, after applying a single row/column operation (like adding all columns to $C_1$), you might find that the determinant is entirely independent of the variable $x$ or $\theta$. Look out for this pattern.