🎯 Key Points
- Matrix multiplication requires columns of A = rows of B; (AB)≠(BA) in general (NOT commutative) — order matters, unlike scalar multiplication
- A⁻¹=adj(A)/det(A) exists ONLY when det(A)≠0 (non-singular); a matrix with det=0 (singular) has NO inverse
- System AX=B: if det(A)≠0, unique solution X=A⁻¹B; if det(A)=0, either NO solution or INFINITELY many — never "exactly one" in the singular case
- Symmetric: A=Aᵗ (mirror across diagonal); Skew-symmetric: A=-Aᵗ (diagonal entries must be 0, since aᵢᵢ=-aᵢᵢ forces aᵢᵢ=0)
Matrices & Determinants
Element aij of matrix A sits at the intersection of row i and column j.
A matrix is a rectangular arrangement of numbers in rows and columns. Matrices are used to represent and solve systems of linear equations and have broad applications in science and engineering.
Types of Matrices
- Row matrix: Only one row
- Column matrix: Only one column
- Square matrix: Equal rows and columns
- Diagonal matrix: Non-zero entries only on main diagonal
- Identity matrix (I): Diagonal with all 1s; AI = IA = A
- Zero matrix: All entries are 0
- Symmetric: A = A^T | Skew-symmetric: A = -A^T
Matrix Operations
A matrix transforms the unit square into a parallelogram by mapping its corner vectors to new positions.
- Addition: Element-wise; matrices must have the same order
- Scalar multiplication: kA multiplies every element by k
- Matrix multiplication: (AB)_ij = row i of A dotted with column j of B; columns of A must equal rows of B
- Transpose (A^T): Rows become columns. (AB)^T = B^T A^T
Determinants
- 2x2: det [[a,b],[c,d]] = ad - bc
- 3x3: Expand along any row/column using cofactors with alternating signs
- det(AB) = det(A) det(B); det(A^T) = det(A); det(kA) = k^n det(A)
- Row of zeros or two identical rows means det = 0
Inverse of a Matrix
A^(-1) = adj(A) / det(A). Exists only when det(A) is not 0.
- Adjoint adj(A): Transpose of the cofactor matrix
System of Linear Equations
- Matrix method: AX = B → X = A^(-1)B when det(A) is not 0
- Cramer's rule: x_i = det(A_i) / det(A)
- Singular (det = 0): either no solution or infinitely many
Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic polynomial p(lambda) = 0, so p(A) = 0. Used to find powers and inverse of A.
Algebra of Matrices: Properties
- Addition (same order only) is commutative A+B=B+A and associative (A+B)+C=A+(B+C); the zero matrix O is the additive identity and −A is the additive inverse.
- Scalar multiplication distributes: k(A+B)=kA+kB and (k+l)A=kA+lA.
- Matrix multiplication is associative (AB)C=A(BC) and distributive A(B+C)=AB+AC, but is NOT commutative in general (AB≠BA); also AB=O does not imply A=O or B=O.
- With a conformable identity matrix, IA=AI=A.
Properties of Transpose
- (A^T)^T = A
- (A+B)^T = A^T + B^T
- (kA)^T = kA^T
- (AB)^T = B^T A^T — note the reversed order (reversal law)
Symmetric and Skew-Symmetric Decomposition
- Symmetric: A = A^T (aij = aji). Skew-symmetric: A = −A^T (aij = −aji), which forces every diagonal entry to be 0.
- Theorem: every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix.
- A = ½(A + A^T) + ½(A − A^T), where ½(A + A^T) is symmetric and ½(A − A^T) is skew-symmetric.
Elementary Operations and Inverse by Elementary Transformations
- Three elementary row (or column) operations: interchange two rows (Ri↔Rj), multiply a row by a non-zero scalar (Ri→kRi), or add a multiple of one row to another (Ri→Ri+kRj).
- To find A⁻¹ by row operations, write A = IA, then reduce the left matrix A to I using row operations applied to both sides; I is transformed into A⁻¹.
- If, during the reduction, an entire row of the left matrix becomes zero, then A is singular and A⁻¹ does not exist.
Invertible Matrices and Uniqueness of Inverse
- A square matrix A is invertible if there exists B with AB = BA = I; then B = A⁻¹.
- The inverse is unique: if B and C are both inverses, then B = BI = B(AC) = (BA)C = IC = C.
- A is invertible ⟺ det(A) ≠ 0 (non-singular).
- Reversal law: (AB)⁻¹ = B⁻¹A⁻¹, and (A^T)⁻¹ = (A⁻¹)^T.
🚀 JEE Advanced Edge
Using Cayley-Hamilton to find high powers or the inverse without direct computation: For a 2×2 matrix A satisfying A²-(trace A)A+(det A)I=0, this single relation lets you express A² (and by repeated substitution, any higher power Aⁿ) as a combination of A and I — far faster than direct repeated matrix multiplication. Multiplying the same relation through by A⁻¹ also gives A⁻¹ directly in terms of A and I when det A≠0, without computing the adjoint matrix at all.
Why every skew-symmetric matrix of odd order has determinant 0: For a skew-symmetric matrix A (Aᵗ=-A) of order n, det(A)=det(Aᵗ)=det(-A)=(-1)ⁿdet(A). If n is ODD, (-1)ⁿ=-1, giving det(A)=-det(A), which forces det(A)=0 — a property that resolves "find the determinant" questions instantly without expanding anything, purely from the matrix's odd order and skew-symmetry.
Worked problem: For A=[[1,2],[2,1]], verify the Cayley-Hamilton theorem and use it to find A⁻¹. Approach: trace(A)=1+1=2, det(A)=1×1-2×2=1-4=-3. Characteristic equation: A²-2A-3I=0, so A²=2A+3I. Rearranging for inverse: from A²-2A-3I=0, divide by A (multiply by A⁻¹): A-2I-3A⁻¹=0, so A⁻¹=(A-2I)/3 = [[1-2,2],[2,1-2]]/3 = [[-1,2],[2,-1]]/3.
Worked Example: Evaluating a 3×3 Determinant
Evaluate det([[1,2,3],[4,5,6],[7,8,9]]).
Expand along Row 1: det = 1·det([[5,6],[8,9]]) − 2·det([[4,6],[7,9]]) + 3·det([[4,5],[7,8]]).
= 1·(45−48) − 2·(36−42) + 3·(32−35) = 1·(−3) − 2·(−6) + 3·(−3) = −3 + 12 − 9 = 0. A determinant of 0 means the matrix is singular (no inverse). Here, each row is an AP with common difference 3, so the rows are linearly dependent.
Worked Example: Solving a System Using Cramer's Rule
Solve 2x + y = 5, x − y = 1 using Cramer's rule.
D = det([[2,1],[1,-1]]) = −2−1 = −3. Dx = det([[5,1],[1,-1]]) = −5−1 = −6. Dy = det([[2,5],[1,1]]) = 2−5 = −3.
x = Dx/D = −6/−3 = 2, y = Dy/D = −3/−3 = 1. Cramer's rule: replace each column of D with the constant column to get the respective numerator determinant.