What is the condition for two matrices to be multiplied?

Matrix multiplication AB is possible (defined) only when the number of columns in A equals the number of rows in B. If A is of order m×n and B is of order n×p, then AB is of order m×p. Example: A is 2×3 and B is 3×4 → AB is 2×4. But BA would require B(3×4) × A(2×3) — 4 columns ≠ 2 rows, so BA is NOT defined.

When does a matrix not have an inverse?

A square matrix A does not have an inverse (is singular) when its determinant |A| = 0. A⁻¹ = adj(A)/|A| — division by zero is undefined. Geometrically, a singular matrix represents a linear transformation that collapses dimensions (e.g., two equations that are parallel — no unique solution).

What is the difference between symmetric and skew-symmetric matrices?

Symmetric matrix: A = Aᵀ (transpose equals itself — elements are symmetric across main diagonal, e.g., aᵢⱼ = aⱼᵢ). Skew-symmetric: A = −Aᵀ (aᵢⱼ = −aⱼᵢ; all diagonal elements = 0). Any square matrix A = ½(A+Aᵀ) + ½(A−Aᵀ) — unique decomposition into symmetric and skew-symmetric parts.

Matrix Multiplication, Transpose and Inverse

Matrices • Class 12 Mathematics • NCERT • CBSE

Matrix multiplication AB is possible only if columns of A = rows of B. If A is m×n and B is n×p, then AB is m×p. Properties: AB ≠ BA (not commutative); A⁻¹ = adj(A)/|A|. For 2×2 matrix [[a,b],[c,d]], |A| = ad−bc, A⁻¹ = (1/|A|)[[d,−b],[−c,a]].

Key Formulas

det([[a,b],[c,d]]) = ad − bc
A⁻¹ = adj(A)/|A|
(AB)ᵀ = BᵀAᵀ
(AB)⁻¹ = B⁻¹A⁻¹
AX = B → X = A⁻¹B

Frequently Asked Questions

What is the condition for two matrices to be multiplied?: Matrix multiplication AB is possible (defined) only when the number of columns in A equals the number of rows in B. If A is of order m×n and B is of order n×p, then AB is of order m×p. Example: A is 2×3 and B is 3×4 → AB is 2×4. But BA would require B(3×4) × A(2×3) — 4 columns ≠ 2 rows, so BA is NOT defined.
When does a matrix not have an inverse?: A square matrix A does not have an inverse (is singular) when its determinant |A| = 0. A⁻¹ = adj(A)/|A| — division by zero is undefined. Geometrically, a singular matrix represents a linear transformation that collapses dimensions (e.g., two equations that are parallel — no unique solution).
What is the difference between symmetric and skew-symmetric matrices?: Symmetric matrix: A = Aᵀ (transpose equals itself — elements are symmetric across main diagonal, e.g., aᵢⱼ = aⱼᵢ). Skew-symmetric: A = −Aᵀ (aᵢⱼ = −aⱼᵢ; all diagonal elements = 0). Any square matrix A = ½(A+Aᵀ) + ½(A−Aᵀ) — unique decomposition into symmetric and skew-symmetric parts.

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