Matrices and Determinants
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Direct answer
Matrix multiplication joins an m × n matrix with an n × p matrix to give an m × p product, and it is not commutative in general. Determinants exist only for square matrices, satisfy |AB| = |A| × |B|, and decide invertibility: A has an inverse exactly when |A| is non-zero, with A^(-1) = adj(A)/|A|. This hinge is what nearly every system-of-equations question turns on.
What you must remember
- Product rules: rows into columns; (AB)^T = B^T A^T; (AB)^(-1) = B^(-1) A^(-1); AB can be zero with neither factor zero, so cancellation laws of arithmetic fail.
- Determinant properties: |A^T| = |A|; |kA| = k^n |A| for an n × n matrix; interchanging two rows flips the sign; a zero row or two proportional rows makes it zero; adding a multiple of one row to another leaves it unchanged.
- Inverse and adjoint: adj(A) is the transpose of the cofactor matrix; A adj(A) = |A| I; A^(-1) = adj(A)/|A|; |adj A| = |A|^(n - 1) for n >= 2.
- Special matrices: symmetric A^T = A; skew-symmetric A^T = -A forces zero diagonal, and every odd-order skew-symmetric matrix is singular; orthogonal A satisfies A A^T = I with |A| = ±1.
- System AX = B with square A: unique solution when |A| is non-zero, given by X = A^(-1) B or Cramer's rule x = D1/D, y = D2/D, z = D3/D; when |A| = 0, the system has no solution if any Di is non-zero and infinitely many if all Di are zero.
- Two-equation reading: |A| = 0 with (adj A) B = 0 means consistent with infinitely many solutions; (adj A) B non-zero means inconsistent.
- Trace: the sum of diagonal entries; tr(A + B) = tr(A) + tr(B) and tr(AB) = tr(BA) even when AB and BA differ.
Common confusion
Two slips dominate. First, |kA| = k|A| — wrong; each of the n rows scales, so the determinant scales by k^n. Second, reading |A| = 0 as "no solution": a singular system may still be consistent with infinitely many solutions, which is exactly why the Di test (or the (adj A) B check) must follow. Add the habit (A + B)^2 = A^2 + 2AB + B^2, valid only when AB = BA, and most lost marks here are accounted for.
Exam-focused takeaway
JEE Main tests determinant evaluation with row operations, adjoint-inverse identities and Cramer-based consistency — mechanical marks if the properties are sharp. JEE Advanced prefers structure: symmetric-skew decomposition A = (A + A^T)/2 + (A - A^T)/2, idempotent and involutory matrices, matrix equations reduced to a polynomial in A, and statements about singular or orthogonal matrices. Translate statements into determinant or transpose algebra before computing anything.
Frequently asked questions
What is |adj A| for an n × n matrix?
It equals |A|^(n - 1) for n >= 2; with A adj(A) = |A| I, this settles most adjoint questions.
When is a linear system solvable?
For square A: uniquely when |A| is non-zero; when |A| = 0, check the Cramer numerators — all zero gives infinitely many solutions, any non-zero gives none.
Is matrix multiplication commutative?
No; AB and BA can differ in value or size, though tr(AB) = tr(BA) always holds when both exist.
How do row operations change a determinant?
Scaling a row multiplies it, swapping two rows changes the sign, and adding a multiple of one row to another leaves it unchanged — the basis of fast evaluation.
What is the inverse of a 2 × 2 matrix?
Swap the diagonal, negate the off-diagonal, divide by ad - bc — valid only when ad - bc is non-zero.