Determinants revision notes

Key Concepts & Definitions

Determinant:

A function that associates every square matrix A=[aij]A = [a_{ij}]A=[aij​] of order nnn with a unique real or complex number. If MMM is the set of square matrices and KKK is the set of numbers, then f:M→Kf: M \to Kf:M→K defined by f(A)=kf(A) = kf(A)=k is the determinant of AAA, denoted by ∣A∣|A|∣A∣, det⁡(A)\det(A)det(A), or Δ\DeltaΔ.

Square Matrix Condition:

Only square matrices possess determinants.

Singular Matrix:

A square matrix AAA is singular if its determinant is zero, i.e., ∣A∣=0|A| = 0∣A∣=0.

Non-Singular Matrix:

A square matrix AAA is non-singular if its determinant is non-zero, i.e., ∣A∣≠0|A| \neq 0∣A∣=0.

Minor (MijM_{ij}Mij​):

The determinant obtained by deleting the ithi^{th}ith row and jthj^{th}jth column of a determinant in which the element aija_{ij}aij​ lies. The minor of an element of a determinant of order n(n≥2)n (n \ge 2)n(n≥2) is a determinant of order n−1n-1n−1.

Cofactor (AijA_{ij}Aij​):

The cofactor of an element aija_{ij}aij​, denoted by AijA_{ij}Aij​, is defined by Aij=(−1)i+jMijA_{ij} = (-1)^{i+j}M_{ij}Aij​=(−1)i+jMij​, where MijM_{ij}Mij​ is the minor of aija_{ij}aij​.

Adjoint of a Matrix (adj A\text{adj } Aadj A):

The transpose of the cofactor matrix [Aij]n×n[A_{ij}]_{n \times n}[Aij​]n×n​ of a square matrix AAA.

Consistent System:

A system of linear equations is consistent if it has one or more solutions.

Inconsistent System:

A system of linear equations is inconsistent if its solution does not exist.

Historical Context:

The Chinese originally used rods on a calculating board for linear equations, resembling determinants. Seki Kowa (1683) had early ideas of determinants. Vandermonde first recognized them as independent functions, Laplace expanded them, Cauchy used the term in its present sense, and Jacobi gave it final acceptance.

Expansion of Determinants

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].

Properties of Determinants

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.

Minors & Cofactors and their Applications

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.

Area of a Triangle & Collinearity

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].

Adjoint & Inverse of a Matrix

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: $|\text{adj } A| = |A|^{n-1}$.

• Advanced Adjoint Properties:
  • $\text{adj}(\text{adj } A) = |A|^{n-2}A$
  • $|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$ → [JEE TIP] Direct formula for tough determinant MCQs.
  • $\text{adj}(AB) = (\text{adj } B)(\text{adj } A)$
  • $\text{adj}(kA) = k^{n-1}\text{adj } A$
  • $\text{adj}(A^T) = (\text{adj } A)^T$ 4. Inverse of a Matrix ($A^{-1}$): A matrix $A$ is invertible if and only if it is non-singular ($|A| \neq 0$). $A^{-1} = \frac{1}{|A|} \text{adj } A$. 5. Product Theorems:
• 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: $\det(A^{-1}) = \frac{1}{\det(A)}$.

System of Linear Equations (Matrix Method & Cramer's Rule)

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 & Integration of Determinants

• 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).

Conditions & Limitations

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.

Previous Year JEE Topics

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.

Memory Aids & JEE Traps

• Trap 1 - Determinant Extraction:
  • Misconception: $|2A| = 2|A|$
  • Correct Understanding: $|2A| = 2^n|A|$ where $n$ is the order of the matrix → [JEE TIP].
• Trap 2 - Alien Cofactors:
  • Misconception: $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = \Delta$
  • Correct Understanding: Elements of one row multiplied by cofactors of another row sums to exactly 0 → [JEE TIP].
• Trap 3 - Transpose Inverse Confusion:
  • Misconception: $(A^T)^{-1}$ and $(A^{-1})^T$ give different matrices.
  • Correct Understanding: The operations of transpose and inverse are commutative: $(A^T)^{-1} = (A^{-1})^T$.
• Trap 4 - Reversal Law Omission:
  • Misconception: $(AB)^{-1} = A^{-1}B^{-1}$
  • Correct Understanding: Matrix inversion over products follows the reversal law: $(AB)^{-1} = B^{-1}A^{-1}$ → [JEE TIP].
• Trap 5 - Meaning of "Non-Trivial Solution":
  • Misconception: A homogeneous system $AX = 0$ having non-trivial solutions means we need to evaluate $X = A^{-1}B$.
  • Correct Understanding: A homogeneous system has non-trivial solutions IF AND ONLY IF $|A| = 0$. You immediately set the determinant to zero. → [JEE TIP]
• Trap 6 - Area Coordinate Given:
  • Misconception: If area of a triangle is $35$, then equating $\frac{1}{2}|...| = 35$ yields the full answer.
  • Correct Understanding: You must equate the determinant to $\pm 35$, which usually generates two valid coordinates for a missing vertex. → [JEE TIP]
• Trap 7 - Adjoint of an Adjoint Determinant:
  • Misconception: $|\text{adj}(\text{adj } A)| = |A|^{n-1}$
  • Correct Understanding: $|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}$. For a $3 \times 3$ matrix, this is $|A|^4$. → [JEE TIP]
• Trap 8 - Cofactor Sign Calculation:
  • Misconception: $A_{ij} = M_{ij}$ directly.
  • Correct Understanding: $A_{ij} = (-1)^{i+j}M_{ij}$. Failing to alternate the signs on the "diamond" positions in a $3 \times 3$ grid is the most common arithmetic error.
• Trap 9 - Singular vs Non-Singular Nomenclature:
  • Misconception: Singular means a single unique solution exists.
  • Correct Understanding: Singular means $|A| = 0$, which implies the inverse does not exist and there is NO unique solution.
• Trap 10 - Evaluating $|-A|$:
  • Misconception: $|-A| = -|A|$ for all matrices.
  • Correct Understanding: $|-A| = (-1)^n |A|$. Thus, $|-A| = |A|$ if $n$ is even, and $-|A|$ if $n$ is odd. → [JEE TIP]