NDA Matrices and Determinants: Operations, Inverse and Linear Equations

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Types of matrices and matrix operations; determinants and basic properties; adjoint and inverse of a square matrix; solution of systems of two or three linear equations by Cramer's Rule and matrix method.

Types of matrices and matrix operations; determinants and basic properties; adjoint and inverse of a square matrix; solution of systems of two or three linear equations by Cramer's Rule and matrix method.

Linear equations, basic algebra and arithmetic with signed numbers.

• Order of a matrix
• Square, diagonal, identity, zero and transpose matrix
• Determinant
• Minor and cofactor
• Adjoint and inverse

• For 2x2 A=[[a,b],[c,d]], det(A)=ad-bc
• A^(-1)=adj(A)/det(A) when det(A)!=0
• For compatible matrices, (AB)_ij = sum a_ik b_kj
• Cramer's Rule: x=Dx/D, y=Dy/D, etc., only when D!=0

• Matrix addition requires the same order.
• AB is defined only when columns of A equal rows of B; AB generally != BA.
• A square matrix has an inverse only when det(A) != 0.
• Cramer's Rule in the standard unique-solution form requires non-zero determinant.

1. For A=[[2,1],[3,2]], det A=4-3=1, so an inverse exists.
2. Solve x+y=5 and 2x-y=1: adding gives 3x=6, x=2, y=3. Matrix or Cramer methods should reproduce the same pair; use the shorter method unless the question specifically tests matrix machinery.
3. If det A=0, do not write adj(A)/0. The inverse does not exist.

• Multiplying matrices element-by-element instead of row-by-column.
• Reversing AB and BA.
• Sign errors in cofactors.
• Attempting inverse when determinant is zero.
• Using Cramer's Rule without checking D.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Check matrix dimensions first. For determinant questions, exploit zeros and simple row / column structure. For linear systems, determine whether the question tests method or just the solution. Under time pressure, avoid expanding a 3x3 determinant blindly if a simplification is available.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.