Matrices and Determinants – JEE Main Maths Formula Sheet & Class 12 Notes | All Formulas, Properties & Results

Types of Matrices — Classification and Key Definitions

Why Matrix Type Classification Is a Direct JEE Main Matrices Question

A matrix A = [aᵢⱼ]ₘₓₙ is a rectangular array of mn numbers arranged in m rows and n columns. The element aᵢⱼ belongs to the iᵗʰ row and jᵗʰ column. All Matrices and Determinants formulas begin with correct identification of matrix type.

Types of matrices in Matrices and Determinants:

Square matrix: m = n (equal rows and columns). The principal diagonal contains elements a₁₁, a₂₂, ..., aₙₙ. Trace of square matrix A = sum of diagonal elements = Σaᵢᵢ = a₁₁ + a₂₂ + ... + aₙₙ. Trace(A+B) = Trace(A) + Trace(B); Trace(kA) = k·Trace(A); Trace(AB) = Trace(BA); Trace(Aᵀ) = Trace(A); Trace(Iₙ) = n.

Diagonal matrix: aᵢⱼ = 0 for i ≠ j (all off-diagonal elements are zero). Written as diag[d₁, d₂, ..., dₙ]. Product of two diagonal matrices = diagonal matrix (element-wise product of diagonals). Determinant of diagonal matrix = product of diagonal elements.

Scalar matrix: diagonal matrix with all diagonal elements equal → aᵢⱼ = K for i = j, 0 for i ≠ j. Equal to KIₙ for some scalar K. A scalar matrix commutes with every square matrix of the same order.

Identity (Unit) matrix Iₙ: aᵢᵢ = 1 and aᵢⱼ = 0 (i ≠ j). Properties: AIₙ = IₙA = A; Iₙ² = Iₙ; Iₙᵏ = Iₙ; |Iₙ| = 1; (Iₙ)⁻¹ = Iₙ.

Triangular matrices: Upper triangular — aᵢⱼ = 0 when i > j (all below-diagonal elements are zero). Lower triangular — aᵢⱼ = 0 when i < j (all above-diagonal elements are zero). Determinant of triangular matrix = product of diagonal elements.

Symmetric matrix: A = Aᵀ (aᵢⱼ = aⱼᵢ for all i, j). The main diagonal can have any values. Sum and product of symmetric matrices: A+B symmetric if A, B symmetric; AB symmetric iff AB = BA (they commute).

Skew-symmetric matrix: A = –Aᵀ (aᵢⱼ = –aⱼᵢ for all i, j). All diagonal elements = 0. Determinant of skew-symmetric matrix of odd order = 0. Every square matrix A can be uniquely written as: A = ½(A + Aᵀ) + ½(A – Aᵀ) = P + Q where P = ½(A+Aᵀ) is symmetric and Q = ½(A–Aᵀ) is skew-symmetric.

Orthogonal matrix: AAᵀ = AᵀA = Iₙ. For orthogonal A: |A| = ±1; A⁻¹ = Aᵀ. Columns (and rows) of orthogonal matrix form orthonormal set.

Special matrices for JEE Main: Idempotent: A² = A. Involutory: A² = I (so A = A⁻¹). Nilpotent: Aᵏ = O for some positive integer k (smallest such k = index of nilpotency). Periodic: Aᵏ⁺¹ = A for some positive integer k. Download the Free PDF for Matrices and Determinants for the complete type table.

Transpose of Matrix — Properties and Formulas

Why Transpose Properties Are Tested in Matrices and Determinants JEE Main Questions

The transpose Aᵀ of an m×n matrix A is the n×m matrix obtained by interchanging rows and columns: if A = [aᵢⱼ]ₘₓₙ then Aᵀ = [aⱼᵢ]ₙₓₘ. Transpose properties in Matrices and Determinants are tested as both symbolic questions and as part of symmetric matrix identification in JEE Main.

Properties of transpose in Matrices and Determinants (JEE Main formulas):

(Aᵀ)ᵀ = A

(A + B)ᵀ = Aᵀ + Bᵀ

(kA)ᵀ = kAᵀ (k is any scalar)

(AB)ᵀ = BᵀAᵀ — note the reversal of order (most important transpose formula in JEE Main)

(ABC)ᵀ = CᵀBᵀAᵀ — reversal extends to any number of factors

|Aᵀ| = |A| — determinant is unchanged by transpose

(A⁻¹)ᵀ = (Aᵀ)⁻¹ — transpose and inverse commute

If A is symmetric: Aᵀ = A. If A is skew-symmetric: Aᵀ = –A.

AA ᵀ and AᵀA are always symmetric matrices (for any matrix A).

Matrix Multiplication — Rules, Properties, and Non-Commutativity

Why Matrix Multiplication Properties Are Core JEE Main Matrices Formulas

Matrix multiplication is the most computationally important operation in Matrices and Determinants. For multiplication AB to be defined: number of columns of A = number of rows of B. If A is m×n and B is n×p, then AB is m×p. The element (AB)ᵢₖ = Σⱼ aᵢⱼ·bⱼₖ (sum over j from 1 to n).

Properties of matrix multiplication in Matrices and Determinants:

Not commutative in general: AB ≠ BA in general (even when both products are defined and same size).

Associative: (AB)C = A(BC).

Distributive: A(B+C) = AB+AC; (A+B)C = AC+BC.

Identity: AIₙ = IₙA = A.

Zero matrix: AO = OA = O.

AB = O does NOT imply A = O or B = O (zero divisors exist in matrix multiplication).

Cancellation fails: AB = AC does NOT imply B = C (unless A is invertible).

If AB = BA (A and B commute), several standard algebraic identities hold: (A+B)² = A²+2AB+B²; (A+B)(A–B) = A²–B², etc.

Determinant product formula: |AB| = |A|·|B| (for square matrices of same order — key Matrices and Determinants formula for JEE Main). Hence |Aⁿ| = |A|ⁿ. |kA| = kⁿ|A| for n×n matrix A.

If A and B are n×n: (AB)⁻¹ = B⁻¹A⁻¹ (inverse reverses order, just like transpose).

Powers of matrices in Matrices and Determinants: A² = A·A; A³ = A²·A; Aⁿ: for idempotent A² = A → Aⁿ = A; for involutory A² = I → A²ⁿ = I, A²ⁿ⁺¹ = A; for nilpotent Aᵏ = O → Aⁿ = O for all n ≥ k. For diagonal matrix diag[d₁,d₂,...,dₙ]: Aⁿ = diag[d₁ⁿ, d₂ⁿ, ..., dₙⁿ]. Download the Free PDF for Matrices and Determinants for matrix multiplication worked examples for JEE Main.

Determinants — Evaluation, Minors, Cofactors, and All Properties

Why Determinant Properties Are the Highest-Frequency Matrices and Determinants JEE Main Section

The determinant of a square matrix A = [aᵢⱼ]ₙₓₙ, written |A| or det(A), is a scalar computed from its elements. For Matrices and Determinants JEE Main, determinant properties are used to simplify evaluation and to classify linear systems.

Determinant formulas in Matrices and Determinants:

2×2: |A| = |a b; c d| = ad – bc.

3×3 expansion along first row: |A| = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ where Cᵢⱼ = (–1)^(i+j)Mᵢⱼ (cofactor). Expansion can be done along any row or column — the value is the same.

Minor Mᵢⱼ: determinant of the submatrix obtained by deleting row i and column j.

Cofactor Cᵢⱼ = (–1)^(i+j) · Mᵢⱼ. Sign pattern: + – +; – + –; + – + for 3×3.

Key identity: sum of products of elements of a row (or column) with corresponding cofactors = |A|. Sum of products of elements of a row with cofactors of ANOTHER row = 0.

Properties of determinants in Matrices and Determinants (all 6 — JEE Main direct questions):

Property I: |A| = |Aᵀ| — value unchanged when rows and columns are interchanged.

Property II: Interchange of any two rows (or columns) multiplies the determinant by –1.

Property III: Multiplying all elements of a single row (or column) by scalar k multiplies the determinant by k. Extracting a common factor k from a row/column: |kRᵢ| = k·|original|. For n×n matrix: |kA| = kⁿ|A|.

Property IV: If two rows (or columns) are identical or proportional, the determinant = 0.

Property V: Adding k times one row to another row leaves the determinant unchanged (Rᵢ → Rᵢ + kRⱼ). Same for columns. This is the key property used to reduce determinants to triangular form.

Property VI: If every element of a row (or column) is a sum of two numbers, the determinant splits into a sum of two determinants. |R₁+R₁', R₂, R₃| = |R₁, R₂, R₃| + |R₁', R₂, R₃|.

Important determinant results in Matrices and Determinants (JEE Main formulas): Determinant of triangular (upper or lower) matrix = product of diagonal elements. |diag[a,b,c]| = abc. |O| = 0 (null matrix). If a row (or column) of a determinant is all zeros, determinant = 0. If Δ(x) = 0 when x = α, then (x–α) is a factor of Δ(x) (factor theorem for determinants). Cofactor determinant: Δ_C (determinant of cofactor matrix) = Δⁿ⁻¹ for n×n matrix with Δ ≠ 0. Product of determinants: |D₁·D₂| = |D₁|·|D₂| (row-by-row or row-by-column multiplication). Download the Free PDF for Matrices and Determinants for all 6 determinant properties with worked examples.

Area of Triangle and Collinearity using Determinants

Why the Area Formula Using Determinant Is a Direct JEE Main Matrices and Determinants Application

One of the most useful geometric applications of determinants in Matrices and Determinants is computing the area of a triangle given its three vertices — bypassing coordinate geometry formulas entirely.

Area of triangle formula using determinant (Matrices and Determinants JEE Main):

For triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):

Area = (1/2)|Δ| where Δ = |x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1|

Expanded: Δ = x₁(y₂–y₃) + x₂(y₃–y₁) + x₃(y₁–y₂)

The modulus is taken because area is always positive. If Δ > 0, vertices are anticlockwise; if Δ < 0, clockwise.

Collinearity condition using determinant (Matrices and Determinants): Three points A, B, C are collinear if and only if area of triangle ABC = 0, i.e.,: |x₁ y₁ 1; x₂ y₂ 1; x₃ y₃ 1| = 0. This is the standard determinant condition for collinearity in Matrices and Determinants and appears regularly in JEE Main coordinate geometry problems solved using matrix methods.

Equation of line through two points using determinant (Matrices and Determinants): The equation of line through (x₁,y₁) and (x₂,y₂) is: |x y 1; x₁ y₁ 1; x₂ y₂ 1| = 0, which expands to the standard two-point form. Download the Free PDF for all Matrices and Determinants geometric applications.

Adjoint and Inverse of Matrix — Complete Formulas for JEE Main

Why Adjoint and Inverse Formulas Are the Most Calculation-Intensive JEE Main Matrices Topic

The adjoint and inverse of a matrix are the two most formula-dense results in Matrices and Determinants and together power the solution of linear systems. Every step has a specific formula — understanding each one is essential for JEE Main maths.

Adjoint of matrix (adj A) in Matrices and Determinants: For a square matrix A = [aᵢⱼ], let Cᵢⱼ = (–1)^(i+j)Mᵢⱼ be the cofactor of aᵢⱼ. The matrix of cofactors is [Cᵢⱼ]. Then adj A = transpose of the cofactor matrix = [Cⱼᵢ] (note: rows and columns interchanged). For 2×2: if A = [a b; c d], then adj A = [d –b; –c a] (swap diagonal elements, negate off-diagonal). For 3×3: compute all 9 cofactors, arrange them in the cofactor matrix, then transpose.

Properties of adjoint in Matrices and Determinants (JEE Main formulas):

A·(adj A) = (adj A)·A = |A|·Iₙ — the most important adjoint property in Matrices and Determinants

|adj A| = |A|^(n–1) for n×n matrix (if |A| ≠ 0)

adj(adj A) = |A|^(n–2)·A for n×n matrix (n ≥ 2, |A| ≠ 0)

|adj(adj A)| = |A|^((n–1)²)

adj(AB) = (adj B)·(adj A) — order reverses, just like transpose and inverse

adj(Aᵀ) = (adj A)ᵀ

adj(kA) = kⁿ⁻¹·(adj A) for n×n matrix and scalar k

If A is symmetric, adj A is also symmetric

If A is diagonal: adj A = diag[|A|/a₁₁, |A|/a₂₂, ..., |A|/aₙₙ] = product of all diag elements / each element

Inverse matrix formula in Matrices and Determinants: A⁻¹ = (adj A) / |A| — this is the master formula for matrix inverse in JEE Main. Condition: A is invertible (has an inverse) if and only if |A| ≠ 0 (non-singular matrix). The inverse of a singular matrix (|A| = 0) does not exist.

Properties of inverse matrix in Matrices and Determinants (JEE Main formulas):

(A⁻¹)⁻¹ = A

(AB)⁻¹ = B⁻¹A⁻¹ — order reverses

(Aᵀ)⁻¹ = (A⁻¹)ᵀ

(kA)⁻¹ = (1/k)A⁻¹ for scalar k ≠ 0

(Aⁿ)⁻¹ = (A⁻¹)ⁿ

|A⁻¹| = 1/|A| = |A|⁻¹

A·A⁻¹ = A⁻¹·A = Iₙ

Iₙ⁻¹ = Iₙ

If A is symmetric, A⁻¹ is also symmetric. If A is orthogonal, A⁻¹ = Aᵀ.

For 2×2: A = [a b; c d], A⁻¹ = [d –b; –c a]/(ad–bc) — direct formula for 2×2 inverse in Matrices and Determinants JEE Main.

Cramer's Rule and System of Linear Equations — All Cases

Why Cramer's Rule Is the Most Concept-Heavy Matrices and Determinants Topic for JEE Main

Cramer's Rule in Matrices and Determinants provides a systematic method for solving and analysing linear systems using determinants. It is one of the most tested sections of Matrices and Determinants in JEE Main maths — both for finding solutions and for determining the type of solution (unique, infinite, or none).

System of 3 linear equations in Matrices and Determinants:

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

Define the determinants for Cramer's Rule:

Δ = |a₁ b₁ c₁; a₂ b₂ c₂; a₃ b₃ c₃| (coefficient determinant — the denominator in Cramer's rule)

Δₓ = |d₁ b₁ c₁; d₂ b₂ c₂; d₃ b₃ c₃| (replace first column by d₁, d₂, d₃)

Δᵧ = |a₁ d₁ c₁; a₂ d₂ c₂; a₃ d₃ c₃| (replace second column by d₁, d₂, d₃)

Δᵤ = |a₁ b₁ d₁; a₂ b₂ d₂; a₃ b₃ d₃| (replace third column by d₁, d₂, d₃)

Cramer's Rule — Three Cases for JEE Main (Non-Homogeneous System):

Case I: Δ ≠ 0 → System is consistent with unique solution: x = Δₓ/Δ, y = Δᵧ/Δ, z = Δᵤ/Δ.

Case II: Δ = 0, but at least one of Δₓ, Δᵧ, Δᵤ is non-zero → System is inconsistent (no solution).

Case III: Δ = 0 AND Δₓ = Δᵧ = Δᵤ = 0 → System has either infinite solutions or no solution (requires further analysis — rank method or back-substitution).

Homogeneous system of linear equations in Matrices and Determinants: a₁x + b₁y + c₁z = 0; a₂x + b₂y + c₂z = 0; a₃x + b₃y + c₃z = 0. x = y = z = 0 (trivial solution) is always a solution, so the system is always consistent. Non-trivial solutions (at least one of x,y,z ≠ 0) exist if and only if Δ = 0. If Δ ≠ 0: only trivial solution exists. If Δ = 0: infinitely many solutions (non-trivial solutions exist).

Matrix method for linear equations in Matrices and Determinants: The system AX = B (where A is the coefficient matrix, X the variable column, B the constant column) has: unique solution X = A⁻¹B if |A| ≠ 0 (A is non-singular → Cramer's rule Case I). No solution or infinite solutions if |A| = 0 (A is singular). The matrix method AX = B is equivalent to Cramer's rule and gives the same classification. Download the Free PDF for all Cramer's rule worked examples for Matrices and Determinants JEE Main.

Important Determinant Results, Identities, and Special Formulas

Why These Matrices and Determinants Formulas Power JEE Main Trick Questions

Beyond the standard evaluation techniques, Matrices and Determinants for JEE Main includes a set of special determinant results and identities that appear as "smart application" questions — where knowing the formula directly gives the answer without full expansion.

Special determinant results in Matrices and Determinants (JEE Main formulas):

Cyclic determinant: |a b c; b c a; c a b| = –(a³+b³+c³–3abc) = –(a+b+c)(a+ωb+ω²c)(a+ω²b+ωc) where ω = cube root of unity.

Vandermonde determinant: |1 a a²; 1 b b²; 1 c c²| = (b–a)(c–a)(c–b).

|a+b a a; a a+b a; a a a+b| = b²(3a+b) — frequently in JEE Main.

|1+a 1 1; 1 1+b 1; 1 1 1+c| = abc(1/a + 1/b + 1/c + 1) = abc + ab + bc + ca.

|a² ab ac; ab b² bc; ac bc c²| = 0 (rows proportional to a:b:c ratios).

Skew-symmetric determinant of odd order = 0.

For anti-symmetric determinant |0 a b; –a 0 c; –b –c 0| = 0 (3×3 skew-symmetric).

|a b c; b+c c+a a+b; a+b+c a+b+c a+b+c| → use R₂→R₂–R₁ and R₃→R₃–R₁ to simplify.

Sum property: a·C_a1 + b·C_b1 + c·C_c1 = |A| (expansion along 1st column). Elements of different row/col times cofactors = 0. Download the Free PDF for Matrices and Determinants for all special determinant identities.

Key matrix determinant results in Matrices and Determinants: |AB| = |A||B|. |A⁻¹| = 1/|A|. |Aⁿ| = |A|ⁿ. |kA| = kⁿ|A| for n×n. |adj A| = |A|^(n–1). If A is orthogonal: |A| = ±1. If A is idempotent (A²=A): |A| = 0 or 1. |A + B| ≠ |A| + |B| in general — common mistake in Matrices and Determinants.