2×2 & 3×3 Matrices · Determinants · Inverses · Multiplication

Matrix Calculator

Compute determinants, matrix inverses, transposes, scalar multiplication, and matrix-matrix products with step-by-step intermediate math.

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Matrix A (2 × 2)

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Matrix B (2 × 2)

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Matrix Multiplication (A × B)

Mathematical Step-by-Step Breakdown:

›[1,1]: Row 1 of A • Col 1 of B = (4 × 1) + (7 × 5) = 39

›[1,2]: Row 1 of A • Col 2 of B = (4 × 3) + (7 × 2) = 26

›[2,1]: Row 2 of A • Col 1 of B = (2 × 1) + (6 × 5) = 32

›[2,2]: Row 2 of A • Col 2 of B = (2 × 3) + (6 × 2) = 18

Comprehensive Guide to Matrix Algebra

Matrix Multiplication (Row × Column)

Unlike ordinary numbers, matrix multiplication is non-commutative: A × B ≠ B × A in general.

Each entry C[i, j] is computed as the dot product between Row i of the first matrix and Column j of the second matrix:

C[i,j] = Σ (A[i,k] × B[k,j])

Determinants & Singular Matrices

The determinant det(A) measures the scaling factor of the linear transformation. Geometrically, it represents how area (2D) or volume (3D) transforms.

If det(A) = 0, the matrix is singular (non-invertible). Its columns are linearly dependent, collapsing spatial dimensions to a lower rank.

Invertibility & Adjugates

The multiplicative inverse satisfies A × A⁻¹ = I, where I is the identity matrix.

The analytical formula uses the transpose of the cofactor matrix (the adjugate matrix) divided by the scalar determinant:

A⁻¹ = (1 / det(A)) × adj(A)