JEEClass 12
Matrices & Determinants Formulas for JEE
All Matrices and Determinants formulas for JEE in one place. Properties, operations, inverse, and system of linear equations.
Matrix Operations
Transpose(AB)ᵀ = BᵀAᵀ
InverseA⁻¹ = adj(A) / |A|
Inverse of product(AB)⁻¹ = B⁻¹A⁻¹
OrthogonalAAᵀ = I (i.e., A⁻¹ = Aᵀ)
IdempotentA² = A
InvolutoryA² = I
Determinant Properties
Scalar multiplication|kA| = kⁿ|A| (for n×n matrix)
Product|AB| = |A| × |B|
Transpose|Aᵀ| = |A|
Inverse|A⁻¹| = 1/|A|
Power|Aⁿ| = |A|ⁿ
Row operationAdding k×(row i) to row j doesn't change |A|
Identical rowsTwo identical rows/cols ⇒ |A| = 0
Adjoint|adj(A)| = |A|ⁿ⁻¹
System of Equations (AX = B)
Unique solution|A| ≠ 0 ⇒ X = A⁻¹B
Cramer's xx = |Dₓ|/|D|, y = |Dᵧ|/|D|, z = |Dᵣ|/|D|
No solution|A| = 0 and (adj A)B ≠ 0
Infinite solutions|A| = 0 and (adj A)B = 0
Homogeneous (AX=0)|A| = 0 ⇒ non-trivial solutions exist
JEE Tips
- Tip 1:Remember |kA| = kⁿ|A| — the power n is the matrix size, not k
- Tip 2:Use row/column operations to simplify determinants before expanding
- Tip 3:For system of equations: check |A| first, then decide the method