Matrices and Determinants formulas for JEE Main & Advanced

23 formula entries for Matrices and Determinants, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Matrix Definition and Order

A = [a_{i j}]_{m \times n}

When to use. When describing or identifying the size of any matrix.

Trap. Confusing rows and columns when writing order; always write rows first.

Types of Matrices

I_{n} = diag (1, 1, \dots, 1)

;

A^{T} = A

for symmetric,

A^{T} = - A

for skew-symmetric.

When to use. When classifying matrices or applying specific properties in problems.

Trap. Assuming all diagonal matrices are scalar; scalar matrices have all diagonal entries equal.

Equality of Matrices

A = B ⟺ a_{i j} = b_{i j}

for all

i, j

When to use. When solving for unknown entries in matrix equations.

Trap. Comparing matrices of different orders; they cannot be equal.

Matrix Addition and Scalar Multiplication

(A + B)_{i j} = a_{i j} + b_{i j}

;

(k A)_{i j} = k \cdot a_{i j}

When to use. When combining matrices or scaling them in linear combinations.

Trap. Adding matrices of different orders; they must be conformable.

Matrix Multiplication

(A B)_{i j} = \sum_{k = 1}^{n} a_{i k} b_{k j}

When to use. When combining linear transformations or solving systems of equations.

Trap. Multiplying in wrong order; matrix multiplication is not commutative ( $A B \neq B A$ generally).

Transpose of a Matrix

(A^{T})_{i j} = a_{j i}

When to use. When dealing with symmetric properties or simplifying expressions.

Trap. Confusing transpose with inverse; they are different.

Determinant of a Matrix

\det (A)

∣ A ∣

; for

2 \times 2

∣ \begin{array}{cc} a & b \\ c & d \end{array} ∣ = a d - b c

When to use. To check if a matrix is invertible (non-zero determinant) or to solve systems via Cramer's rule.

Trap. Applying determinant formula to non-square matrices; only square matrices have determinants.

Properties of Determinants

\det (A B) = \det (A) \det (B)

;

\det (k A) = k^{n} \det (A)

When to use. When simplifying determinant expressions or proving identities.

Trap. Assuming $\det (A + B) = \det (A) + \det (B)$ ; this is false.

Adjoint of a Matrix

adj (A) = (C_{i j})^{T}

where

C_{i j}

is cofactor of

a_{i j}

When to use. When finding inverse or solving matrix equations using $A^{- 1} = \frac{adj (A)}{\det (A)}$ .

Trap. Confusing adjoint with inverse; adjoint is used to compute inverse.

Inverse of a Matrix

A^{- 1} = \frac{1}{\det (A)} adj (A)

When to use. To solve $A X = B$ as $X = A^{- 1} B$ , or to verify matrix properties.

Trap. Assuming every square matrix has an inverse; only non-singular matrices do.

Solving Linear Systems Using Matrices

X = A^{- 1} B

; Cramer's rule:

x_{i} = \frac{\det (A_{i})}{\det (A)}

When to use. When solving linear equations with same number of equations as unknowns.

Trap. Applying Cramer's rule when $\det (A) = 0$ ; system may be inconsistent or have infinite solutions.

Rank of a Matrix

rank (A) = number of non-zero rows in row echelon form

When to use. To determine consistency of linear systems or linear independence.

Trap. Confusing rank with determinant; a non-zero determinant implies full rank, but zero determinant doesn't mean rank 0.

Elementary Row Operations

R_{i} \leftrightarrow R_{j}

R_{i} \to k R_{i}

R_{i} \to R_{i} + k R_{j}

When to use. When computing inverse via Gauss-Jordan or solving systems.

Trap. Using column operations when row operations are specified; they are different.

Symmetric and Skew-Symmetric Matrices

A^{T} = A

(symmetric);

A^{T} = - A

(skew-symmetric)

When to use. When decomposing a matrix into symmetric and skew-symmetric parts: $A = \frac{A + A^{T}}{2} + \frac{A - A^{T}}{2}$ .

Trap. Assuming a symmetric matrix must be diagonal; it only needs symmetry about diagonal.

Cramer's Rule

x_{i} = \frac{\det (A_{i})}{\det (A)}

When to use. When solving small systems (2 or 3 variables) with unique solution.

Trap. Applying when $\det (A) = 0$ ; rule fails.

Minor and Cofactor

C_{i j} = (- 1)^{i + j} M_{i j}

When to use. When computing determinant via expansion or adjoint.

Trap. Forgetting the sign factor $(- 1)^{i + j}$ when computing cofactor.

Properties of Transpose

(A B)^{T} = B^{T} A^{T}

When to use. When simplifying expressions involving transposes.

Trap. Assuming $(A B)^{T} = A^{T} B^{T}$ ; order reverses.

Singular and Non-Singular Matrices

\det (A) = 0 ⟹

singular;

\det (A) \neq 0 ⟹

non-singular

When to use. To check invertibility or existence of unique solutions.

Trap. Assuming a zero matrix is the only singular matrix; many non-zero matrices can be singular.

Trace of a Matrix

tr (A) = \sum_{i = 1}^{n} a_{i i}

When to use. When dealing with eigenvalues or matrix invariants.

Trap. Applying trace to non-square matrices; only defined for square matrices.

Orthogonal Matrix

A^{T} A = I

When to use. When dealing with rotations or preserving lengths.

Trap. Assuming $\det A = + 1$ for an orthogonal matrix; it is $\pm 1$ , and both rows and columns must be orthonormal.

Matrix Polynomials

p (A) = a_{n} A^{n} + \dots + a_{0} I

; Cayley-Hamilton: if

p (λ) = \det (A - λ I)

then

p (A) = O

When to use. To compute high powers of matrices or find inverse using polynomial.

Trap. Assuming scalar polynomial properties hold for matrices; multiplication order matters.

Characteristic Equation and Eigenvalues

\det (A - λ I) = 0

When to use. To find eigenvalues, which are used in diagonalization and stability analysis.

Trap. Confusing characteristic equation with $\det (A) = 0$ ; they are different.

Cayley-Hamilton Theorem

A^{n} + c_{n - 1} A^{n - 1} + \dots + c_{0} I = 0

When to use. To compute $A^{- 1}$ or high powers of $A$ without diagonalization.

Trap. Applying to non-square matrices; only square matrices.

Read the full chapter: Matrices and Determinants for JEE, with what JEE asks and the common traps.