Matrices & Determinants Calculator

Addition, Subtraction, Multiplication, Determinant & Inverse (2×2 and 3×3)

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Matrix A (m×n) × Matrix B (n×p). Dimensions must match.

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Quick Answer – What are Matrices & Determinants?

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix operations include addition, subtraction, and multiplication. The determinant is a scalar value computed from a square matrix, indicating invertibility (non-zero determinant ⇒ invertible). The inverse matrix A−1 satisfies A A−1 = I.

Key Formulas

2×2 Determinant: |a b; c d| = ad − bc
2×2 Inverse: A−1 = (1/det) [[d, −b], [−c, a]]
3×3 Determinant: Expansion by minors
Matrix multiplication: Cij = Σ Aik Bkj

Solved Examples

Example 1: Determinant of [[2,3],[1,4]]

det = 2×4 − 3×1 = 8 − 3 = 5

Example 2: Inverse of [[4,7],[2,6]]

det = 4×6 − 7×2 = 24 − 14 = 10
Inverse = (1/10) [[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]

Example 3: Multiply [[1,2],[3,4]] × [[5,6],[7,8]]

Result = [[1×5+2×7, 1×6+2×8], [3×5+4×7, 3×6+4×8]] = [[19,22],[43,50]]

Frequently Asked Questions

Q1. Can I use decimal or negative numbers?
Yes, all real numbers are accepted. Use a minus sign for negatives and a dot for decimals. Commas are automatically converted to dots.
Q2. Why does matrix multiplication require compatible dimensions?
The number of columns of the first matrix must equal the number of rows of the second. This tool allows you to choose the inner dimension; if it doesn't match, an error is shown.
Q3. What happens if the determinant is zero for inverse?
The matrix is singular and does not have an inverse. The calculator will display "Matrix is singular, inverse does not exist".
Q4. Can I compute determinant of a non-square matrix?
Determinant is defined only for square matrices. The tool restricts to 2×2 and 3×3.
Q5. How precise are the results?
Results are displayed to 4 decimal places for clarity, using standard floating-point arithmetic.