Matrices and Determinants
What the exam tests
Matrices and determinants in the Tamil Nadu HSE Business Mathematics paper test whether you can classify matrices, carry out addition, multiplication and transposition accurately, evaluate determinants of order 2 and 3, and use the adjoint method or Cramer's rule to solve small linear systems. Questions are usually computational and award method marks, so show every cofactor.
Core rules
- A matrix is singular when its determinant is zero; only non-singular matrices have inverses.
- For A = [[a,b],[c,d]], det A = ad − bc and A⁻¹ = (1/det A)[[d,−b],[−c,a]].
- The adjoint is the transpose of the cofactor matrix: adj A = Cᵀ. Never forget the transpose.
- det(AB) = det A · det B, but AB ≠ BA in general.
- Cramer's rule: x = Dₓ/D, y = D_y/D, provided D ≠ 0.
Worked example
Evaluate |1 2 3; 0 1 4; 5 6 0| by expanding along row 1. = 1(1·0 − 4·6) − 2(0·0 − 4·5) + 3(0·6 − 1·5) = 1(−24) − 2(−20) + 3(−5) = −24 + 40 − 15 = 1. Because the determinant is 1 ≠ 0, the matrix is non-singular and its inverse is adj A / 1.
Solve 2x + 3y = 13, x + 2y = 8 by Cramer's rule. D = 4 − 3 = 1; Dₓ = 26 − 24 = 2; D_y = 16 − 13 = 3. So x = 2, y = 3.
Traps
- Sign errors in the alternating + − + pattern of cofactor expansion.
- Writing adj A instead of (adj A)/det A for the inverse.
- Using Cramer's rule when D = 0; then the system has no unique solution.
- Assuming AB = BA or that (AB)ᵀ = AᵀBᵀ; the correct rule is BᵀAᵀ.