Matrices and Determinants
Matrices and Determinants is the algebra engine of the paper: matrix order and equality, elementary row operations, determinant evaluation, adjoint and inverse, and solving a linear system by the matrix method. Questions regularly combine a 3 × 3 determinant with an inverse or a consistency check.
Rules to keep straight
- Matrix multiplication requires columns of A to equal rows of B. AB ≠ BA, and AB = 0 does not force A = 0 or B = 0.
- (AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹ - the order reverses.
- |AB| = |A||B|; |kA| = kⁿ|A| for an n × n matrix; |adj A| = |A|ⁿ⁻¹.
- A⁻¹ = adj A/|A|, valid only when |A| ≠ 0 (non-singular).
- A square matrix is invertible ⇔ |A| ≠ 0 ⇔ its rows and columns are linearly independent.
- A skew-symmetric matrix of odd order has determinant 0.
Worked example
For A = [[1, 2], [3, 4]]: |A| = 1(4) − 2(3) = −2. The adjoint swaps the diagonal entries and negates the off-diagonal ones, so adj A = [[4, −2], [−3, 1]]. Hence A⁻¹ = (−1/2)[[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]]. Check: |adj A| = 4 − 6 = −2 = |A|^(2−1), matching the formula.
Traps
- Writing (A + B)² = A² + 2AB + B², which fails because AB ≠ BA.
- Using |kA| = k|A| instead of kⁿ|A|.
- Applying adj A/|A| when |A| = 0.
- Interchanging rows in a determinant without flipping the sign, or applying a column operation to only one entry.