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📑 In This Chapter Guide (Table of Contents)
1. Types of Matrices & Multiplication Rules
A matrix is an ordered rectangular array of numbers or functions with m rows and n columns (order m × n):
- Symmetric Matrix: A square matrix
Ais symmetric ifAᵀ = A(i.e.,a_ij = a_jifor all i, j). - Skew-Symmetric Matrix: A square matrix
Ais skew-symmetric ifAᵀ = -A(i.e.,a_ij = -a_ji). All principal diagonal elements of a skew-symmetric matrix are strictly zero. - Theorem: Any square matrix
Acan be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix:A = ½ (A + Aᵀ) [Symmetric] + ½ (A - Aᵀ) [Skew-Symmetric]. - Matrix Multiplication Condition: Product
ABis defined if and only if the number of columns in A equals the number of rows in B. Matrix multiplication is generally non-commutative (AB ≠ BA).
2. Determinants, Minors, Cofactors & Adjoint
A determinant is a scalar value associated with any square matrix. Determinant of transpose: |Aᵀ| = |A|.
Cofactor (A_ij): A_ij = (-1)^(i + j) · M_ij (where M_ij is the minor obtained by deleting row i and column j).
Adjoint of a Square Matrix (adj A): The transpose of the matrix of cofactors: adj A = [A_ij]ᵀ.
Fundamental Adjoint Theorems (High-Yield 1-Mark Formulae):
A · (adj A) = (adj A) · A = |A| · I- For an n × n matrix:
|adj A| = |A|^(n - 1) |adj (adj A)| = |A|^((n - 1)²)adj (AB) = (adj B) · (adj A)
3. Inverse of a Matrix & Matrix Method for Linear Systems
A square matrix A is invertible (non-singular) if and only if |A| ≠ 0. The inverse matrix is:
A⁻¹ = (1 / |A|) · adj(A)
Solving System of Linear Equations (AX = B):
For system: a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃:
1. Form coefficient matrix A = [[a₁, b₁, c₁], [a₂, b₂, c₂], [a₃, b₃, c₃]], variable vector X = [[x], [y], [z]], and constant vector B = [[d₁], [d₂], [d₃]].
2. Calculate determinant |A|. If |A| ≠ 0, the system is consistent with a unique solution:
X = A⁻¹ · B
💡 Frequently Asked Questions (FAQ)
❓ If A is a square matrix of order 3 and |A| = 5, find the value of |adj A|.
Using the formula |adj A| = |A|^(n - 1): here n = 3 and |A| = 5. Therefore: |adj A| = 5^(3 - 1) = 5² = 25.
❓ What is the condition for a system AX = B to have infinitely many solutions versus no solution when |A| = 0?
When |A| = 0 (singular matrix): Calculate (adj A) · B. (1) If (adj A) · B ≠ O (zero matrix), the system is inconsistent and has NO SOLUTION; (2) If (adj A) · B = O, the system may be consistent with INFINITELY MANY SOLUTIONS or inconsistent.
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