Determinants revision notes

1. Determinant of Order 1: For $A = [a_{11}]$, $|A| = a_{11}$. 2. Determinant of Order 2: For $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, $\Delta = a_{11}a_{22} - a_{21}a_{12}$. 3. Determinant of Order 3: Can be expanded along any of its 3 rows or 3 columns. The expansion along the first row ($R_1$) is: $|A| = a_{11}(a_{22}a_{33} - a_{32}a_{23}) - a_{12}(a_{21}a_{33} - a_{31}a_{23}) + a_{13}(a_{21}a_{32} - a_{31}a_{22})$.

• Expanding along any row or column yields the same value.
• Calculation trick: Always expand along the row or column containing the maximum number of zeros JEE Tip.

1. Reflection Property: The determinant remains unaltered if its rows are changed into columns and columns into rows ($|A| = |A^T|$). 2. Switching Property: If any two rows (or columns) are interchanged, the sign of the determinant changes. 3. Repetition Property: If any two rows (or columns) of a determinant are identical (or proportional), the value of the determinant is zero. JEE Tip Always check for proportional rows before expanding manually to save time. 4. Scalar Multiple Property: If each element of a row (or column) is multiplied by a constant $k$, the value of the determinant is multiplied by $k$.

• As a corollary, if $A$ is a square matrix of order $n$, then $|kA| = k^n|A|$. 5. Sum Property: If some or all elements of a row/column are expressed as the sum of two (or more) terms, the determinant can be expressed as the sum of two (or more) determinants. 6. Invariance Property: The value of a determinant remains same if we apply the operation $R_i \to R_i + kR_j$ or $C_i \to C_i + kC_j$. 7. Product of Determinants: Multiplication can be done row-by-row or row-by-column.

1. Determinant Value via Cofactors: The determinant $\Delta$ is equal to the sum of the products of elements of any row (or column) with their corresponding cofactors.

• $\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$. 2. Alien Cofactor Property: If elements of a row (or column) are multiplied with the cofactors of any other row (or column), their sum is strictly zero.
• $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0$. JEE Tip This is frequently tested in matrix-matching questions.

The area of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is: $\Delta = \frac{1}{2} \left| \begin{matrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{matrix} \right|$

• Collinearity Condition: Three points are collinear if and only if the area of the triangle formed by them is zero ($\Delta = 0$).
• Equation of a Line: The equation of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$ is obtained by equating the area of the triangle formed by $(x, y), (x_1, y_1)$, and $(x_2, y_2)$ to zero.
• Absolute Value: Area is always a positive quantity, so take the absolute value of the determinant. If the area is given in a problem, both positive and negative values of the determinant must be used for finding unknown coordinates JEE Tip.

1. Adjoint of a $2 \times 2$ Matrix: For $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$, $\text{adj } A$ is found directly by interchanging $a_{11}$ and $a_{22}$, and changing the signs of $a_{12}$ and $a_{21}$. JEE Tip 2. Fundamental Theorem of Adjoint: For any square matrix $A$ of order $n$: $A(\text{adj } A) = (\text{adj } A)A = |A|I_n$. 3. Determinant of Adjoint: Taking the determinant on both sides of the above equation yields: $|\text{adj } A| = |A|^{n-1}$.

• If $A, B$ are non-singular of the same order, $AB$ and $BA$ are non-singular.
• $|AB| = |A||B|$.
• Reversal Law of Inverse: $(AB)^{-1} = B^{-1}A^{-1}$.
• Determinant of Inverse: $\det(A^{-1}) = \frac{1}{\det(A)}$.

A system of equations: $a_1x + b_1y + c_1z = d_1$ $a_2x + b_2y + c_2z = d_2$ $a_3x + b_3y + c_3z = d_3$ can be expressed as $AX = B$, where $A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, $B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}$.

Condition 1: $A$ is non-singular ($|A| \neq 0$)

• The system has a unique solution given by $X = A^{-1}B$.
• The system is consistent.

Condition 2: $A$ is singular ($|A| = 0$)

In this case, calculate $(\text{adj } A)B$.

• If $(\text{adj } A)B \neq O$ (Zero matrix): The system has no solution and is inconsistent.
• If $(\text{adj } A)B = O$: The system may be either consistent (infinitely many solutions) or inconsistent (no solution) depending on further geometric intersection conditions of the planes.

Cramer's Rule [JEE Advanced]

Let $\Delta = |A|$. Let $\Delta_1, \Delta_2, \Delta_3$ be determinants obtained by replacing $C_1, C_2, C_3$ of $\Delta$ with column matrix $B$ respectively.

• $x = \frac{\Delta_1}{\Delta}, y = \frac{\Delta_2}{\Delta}, z = \frac{\Delta_3}{\Delta}$
• If $\Delta \neq 0$: Unique solution.
• If $\Delta = 0$ and at least one of $\Delta_1, \Delta_2, \Delta_3 \neq 0$: No solution.
• If $\Delta = \Delta_1 = \Delta_2 = \Delta_3 = 0$: Usually infinitely many solutions.

Homogeneous System of Equations ($d_1 = d_2 = d_3 = 0$) [JEE Advanced]

• If $|A| \neq 0$, the system has only the trivial solution ($x = 0, y = 0, z = 0$).
• If $|A| = 0$, the system has non-trivial solutions (infinitely many solutions). JEE Tip Standard trigger in JEE Advanced: "System has a non-zero solution" immediately implies set $|A| = 0$.

• Differentiation: To differentiate a determinant $\Delta(x)$ with respect to $x$, differentiate one row (or column) at a time while keeping the others unchanged, and add the resulting determinants.
• Integration: If the elements of only one row (or column) are functions of $x$ and the rest are constants, the integral of the determinant is the determinant of the integrals of that specific row (or column).

1. Applicability of Determinants: Determinants are only defined for square matrices (where number of rows = number of columns).
2. Inverse Existence: The inverse of a matrix $A$ exists if and only if $A$ is non-singular ($|A| \neq 0$). If a system of equations yields $|A| = 0$, Matrix Inversion Method $X = A^{-1}B$ CANNOT be used to find a unique solution.
3. Area Evaluation: The determinant formula for the area of a triangle yields a signed algebraic value. For pure area (geometry), the absolute value must be taken. However, when the area is given to find a variable coordinate, both $+ \Delta$ and $- \Delta$ bounds must be solved.
4. Limits on $(\text{adj } A)B = O$: If $|A| = 0$ and $(\text{adj } A)B = O$, the system may have infinitely many solutions, but it can also have no solution (if the planes represent parallel identical lines but distinct 3D planes). Further checking using 3D geometry is strictly required.

• Modulus vs. Determinant Notation: The notation $|A|$ is read as "determinant of A", not "modulus of A". Do not mistakenly assume $|A|$ is strictly positive.
• Constant Extraction Misconception: In matrices, $kA$ implies multiplying every element by $k$. In determinants, taking $k$ common only extracts it from one row or column. Hence, for an $n \times n$ matrix, $|kA| = k^n|A|$, not $k|A|$.
• Alien Cofactor Zero Check: Students often calculate $\sum a_{1j}A_{2j}$ mistakenly expecting the determinant value $\Delta$. Remember, summing elements with cofactors of a different row always evaluates precisely to $0$.
• Adjoint Sign Errors: While directly writing the Adjoint of a $2 \times 2$ matrix, students swap the main diagonal but forget to flip the signs on the non-diagonal elements (or vice versa). $\text{adj} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
• Polynomial Roots via Determinant: When solving determinant equations like $|A(x)| = 0$ to find $x$, check if dropping/canceling a factor like $(x-1)$ across rows lost a root. Always take the factor out common, leaving it as $(x-1)|...| = 0$.

1. Properties of Determinants: Heavily tested to simplify complex $3 \times 3$ and general $n \times n$ matrices before expansion.
2. System of Linear Equations (Consistency): The conditions for unique, infinite, and zero solutions using Cramer's rule ($\Delta, \Delta_1, \Delta_2, \Delta_3$) and the Adjoint method. Most common Advanced topic: Homogeneous systems with non-trivial solutions ($\Delta = 0$).
3. Adjoint Properties: Direct use of $|\text{adj } A| = |A|^{n-1}$ and $\text{adj}(\text{adj } A) = |A|^{n-2}A$.
4. Polynomials derived from Determinants: Finding maximum or minimum values of functions where the function is defined as a $3 \times 3$ determinant involving trigonometric terms.