Math · Algebra

Determinants formulas for JEE

Every Determinants formula you need for JEE, grouped by concept.

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23 formulas4 concepts
01Determinants02Properties of Determinants03Adjoint and Inverse of a Matrix04System of Linear Equations
01

Determinants

5 formulas

Area of a Triangle using Determinants

Δ=12∣x1y11x2y21x3y31∣\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|

Calculates the algebraic area of a triangle given its three vertices. Take the absolute value for geometric area.

applies whenVertices are (x1, y1), (x2, y2), and (x3, y3).
areatrianglecoordinate-geometry

Determinant of 3x3 Matrix

Δ=a11(a22a33−a32a23)−a12(a21a33−a31a23)+a13(a21a32−a31a22)\Delta = 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})

Standard expansion of a 3x3 determinant along the first row.

applies whenMatrix must be a 3x3 square matrix.
determinant3x3expansion

Determinant Expansion via Cofactors

Δ=ai1Ai1+ai2Ai2+ai3Ai3\Delta = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}

Value of determinant as the sum of product of elements of any row/col with their corresponding cofactors.

applies whenExpansion along the i-th row of a 3x3 determinant.
expansioncofactor

Equation of Line using Determinants

∣xy1x1y11x2y21∣=0\left| \begin{matrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{matrix} \right| = 0

Equation of a straight line passing through two points.

applies whenPoints are (x1, y1) and (x2, y2).
linecollinearcoordinate-geometry

Determinant of 2x2 Matrix

Δ=a11a22−a12a21\Delta = a_{11}a_{22} - a_{12}a_{21}

Calculates the determinant of a 2x2 square matrix.

applies whenMatrix must be a 2x2 square matrix.
determinant2x2
02

Properties of Determinants

4 formulas

Alien Cofactor Property

ai1Ak1+ai2Ak2+ai3Ak3=0a_{i1}A_{k1} + a_{i2}A_{k2} + a_{i3}A_{k3} = 0

Sum of products of elements of one row with cofactors of a different row is zero.

applies wheni \neq k
cofactorpropertyzero

Differentiation of a Determinant

ddx∣f(x)g(x)u(x)v(x)∣=∣f′(x)g′(x)u(x)v(x)∣+∣f(x)g(x)u′(x)v′(x)∣\frac{d}{dx} \left| \begin{matrix} f(x) & g(x) \\ u(x) & v(x) \end{matrix} \right| = \left| \begin{matrix} f'(x) & g'(x) \\ u(x) & v(x) \end{matrix} \right| + \left| \begin{matrix} f(x) & g(x) \\ u'(x) & v'(x) \end{matrix} \right|

Differentiating a determinant by taking the derivative of one row (or column) at a time.

applies whenElements of the determinant must be differentiable functions of x.
differentiationcalculusjee-advanced

Determinant of Matrix Product

∣AB∣=∣A∣∣B∣|AB| = |A||B|

The determinant of a product of matrices is the product of their determinants.

applies whenA and B must be square matrices of the same order.
productdeterminantproperty

Determinant of Scalar Multiple

∣kA∣=kn∣A∣|kA| = k^n|A|

Determinant of a matrix multiplied by a scalar k.

applies whenA is a square matrix of order n.
determinantscalarproperty
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03

Adjoint and Inverse of a Matrix

11 formulas

Adjoint of an Adjoint

adj(adj A)=∣A∣n−2A\text{adj}(\text{adj } A) = |A|^{n-2}A

Formula evaluating the adjoint of an adjoint of a matrix.

applies whenA is a non-singular square matrix of order n \geq 2.
adjointhigher-orderjee-advanced

Fundamental Theorem of Adjoint

A(adj A)=(adj A)A=∣A∣IA(\text{adj } A) = (\text{adj } A)A = |A|I

The product of a square matrix and its adjoint yields a scalar matrix of its determinant.

applies whenA is a square matrix of order n.
adjointidentity

Adjoint of a Scalar Multiple

adj(kA)=kn−1adj A\text{adj}(kA) = k^{n-1}\text{adj } A

Formula for taking a scalar multiplier out of an adjoint operator.

applies whenA is a square matrix of order n, k is a scalar.
adjointscalarjee-advanced

Cofactor of an Element

Aij=(−1)i+jMijA_{ij} = (-1)^{i+j}M_{ij}

Relationship between the cofactor and the minor of an element.

applies whenM_ij is the minor of element a_ij.
cofactorminor

Determinant of Adjoint Matrix

∣adj A∣=∣A∣n−1|\text{adj } A| = |A|^{n-1}

Formula for the determinant of the adjoint of A.

applies whenA is a square matrix of order n.
adjointdeterminant

Determinant of Adjoint of Adjoint

∣adj(adj A)∣=∣A∣(n−1)2|\text{adj}(\text{adj } A)| = |A|^{(n-1)^2}

Formula for the determinant of the adjoint of the adjoint of A.

applies whenA is a square matrix of order n.
determinantadjointjee-advanced

Inverse of Adjoint Identity

(adj A)−1=adj (A−1)(\text{adj } A)^{-1} = \text{adj } (A^{-1})

The inverse of an adjoint matrix is equal to the adjoint of the inverse matrix.

applies whenA must be a non-singular matrix.
adjointinverseidentity

Inverse of a Matrix

A−1=1∣A∣adj AA^{-1} = \frac{1}{|A|} \text{adj } A

Formula to compute the inverse of a matrix using its adjoint.

applies whenMatrix A must be non-singular (|A| \neq 0).
inverseadjoint

Double Inverse Property

(A−1)−1=A(A^{-1})^{-1} = A

The inverse of an inverse matrix is the original matrix.

applies whenA must be a non-singular matrix.
inverseproperty

Inverse from Polynomial Equation

A−1=−1b(A+aI)A^{-1} = -\frac{1}{b}(A + aI)

Finding the inverse using a characteristic/annihilating polynomial equation.

applies whenDerived from A^2 + aA + bI = O with b \neq 0.
inversepolynomialequation

Reversal Law for Matrix Inverse

(AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}

The inverse of a product of matrices is the product of their inverses in reverse order.

applies whenA and B must be non-singular square matrices of the same order.
inverseproductreversal-law
04

System of Linear Equations

3 formulas

Cramer's Rule

x=Δ1Δ,y=Δ2Δ,z=Δ3Δx = \frac{\Delta_1}{\Delta}, \quad y = \frac{\Delta_2}{\Delta}, \quad z = \frac{\Delta_3}{\Delta}

Method to solve a system of linear equations using determinants.

applies whenDeterminant of coefficient matrix \Delta \neq 0.
cramers-rulesystem-of-equationsjee-advanced

Condition for Inconsistency

∣A∣=0 and (adj A)B≠O|A| = 0 \text{ and } (\text{adj } A)B \neq O

Condition indicating that a system of linear equations has no solution.

applies whenEvaluated when matrix A is singular.
system-of-equationsinconsistentno-solution

Matrix Solution to Linear Systems

X=A−1BX = A^{-1}B

Unique solution to the matrix equation AX = B.

applies whenA is a non-singular matrix (|A| \neq 0).
system-of-equationsinversesolution
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