Matrix Determinant/Inverse & Binary/Hex Radix Calculator

Calculate matrix determinant, trace, and inverse alongside Binary, Hexadecimal, and Octal radix conversions.

Matrix Determinant (det A) & Binary/Hex Radix Representation
det(A) = 14 • 0xFF (11111111₂)
Matrix Inverse A⁻¹: [0.357, -0.143; -0.214, 0.286] • Trace(A) = 9
Determinant (ad − bc) 14
Inverse Matrix A⁻¹ [0.357, -0.143; -0.214, 0.286]
Binary (Base-2 of 255) 0b11111111
Hexadecimal (Base-16) / Octal 0xFF / 0o377
Inputs
Row 1, Column 1.
Row 1, Column 2.
Row 2, Column 1.
Row 2, Column 2.
Non-negative integer to convert across bases.
Results & Graph
Matrix Determinant (det A) & Binary/Hex Radix Representation
det(A) = 14 • 0xFF (11111111₂)
Matrix Inverse A⁻¹: [0.357, -0.143; -0.214, 0.286] • Trace(A) = 9
Determinant (ad − bc)
14
Inverse Matrix A⁻¹
[0.357, -0.143; -0.214, 0.286]
Binary (Base-2 of 255)
0b11111111
Hexadecimal (Base-16) / Octal
0xFF / 0o377

Matrix Invariants: |Determinant| vs. |Trace|

Comparison of matrix determinant and diagonal trace magnitude
Graph 1 • Parallelogram Spanned by Matrix Column Vectors Area of transformed unit square equals |det(A)|
2D Linear Transformation of Unit Square • Column Vector 1: (4, 3) • Column Vector 2: (2, 5) • Parallelogram Area |det(A)| = 14.00 • Trace tr(A) = a₁₁ + a₂₂ = 9.00
Graph 2 • 8-Bit Binary Register & Positional Powers of 2 Shows active bits (2⁷ down to 2⁰), Hex, and Octal representations
1 2^7 (128) 1 2^6 (64) 1 2^5 (32) 1 2^4 (16) 1 2^3 (8) 1 2^2 (4) 1 2^1 (2) 1 2^0 (1) Decimal: 255 • Binary: 0b11111111 • Hex: 0xFF • Octal: 0o377
Graph 3 • Determinant, Trace & Frobenius Norm Comparison Key linear algebra invariants of Matrix A
Determinant |det(A)| 14.00 Matrix Trace tr(A) = a₁₁ + a₂₂ 9.00 Frobenius Norm √(Σ a_ij²) 7.35
Summary: Matrix determinant is 14 and 255 in binary is 11111111.
Summary: The 2×2 matrix [[4, 2], [3, 5]] has determinant det(A) = 14, trace = 9, and inverse [0.357, -0.143; -0.214, 0.286]. Converting decimal 255 across bases yields binary 0b11111111, hexadecimal 0xFF, and octal 0o377.
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Formula & Mathematical Methodology: 2×2 Matrix Cramer Inversion & Positional Radix Expansion

det(A) = a11a22 − a12a21,   A−1 = 1det(A) a22−a12−a21a11
det(A) = a₁₁a₂₂ − a₁₂a₂₁; A⁻¹ = (1 / det(A)) × [a₂₂, −a₁₂; −a₂₁, a₁₁]
  • det(A): Area scaling factor of the linear transformation (invertible iff det(A) ≠ 0)

Rounding: Matrix inverse entries round to 3 decimal places.

Worked Example: Matrix [[4, 2], [3, 5]] and Radix Conversion of 255

Find determinant and inverse of [[4, 2], [3, 5]] and convert 255 to binary and hex.

  1. det(A) = (4 × 5) − (2 × 3) = 20 − 6 = 14; Trace = 4 + 5 = 9.
  2. Inverse A⁻¹ = (1/14) × [[5, −2], [−3, 4]] = [[0.357, −0.143], [−0.214, 0.286]].
  3. 255 in binary = 0b11111111, hex = 0xFF, octal = 0o377.
Result: det(A) = 14, Trace = 9, and 255 = 0xFF (11111111₂).

Frequently Asked Questions

When does a square matrix fail to have an inverse?

A matrix is singular (non-invertible) when its determinant equals zero, meaning its rows or columns are linearly dependent.

Mathematics & Statistics · Linear Algebra & Discrete Math

Matrix Determinant/Inverse & Binary/Hex Radix Calculator: inputs, method, and how to read the result

Calculate matrix determinant, trace, and inverse alongside Binary, Hexadecimal, and Octal radix conversions.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Matrix Entry a₁₁ (Top-Left): Row 1, Column 1.
  • Matrix Entry a₁₂ (Top-Right): Row 1, Column 2.
  • Matrix Entry a₂₁ (Bottom-Left): Row 2, Column 1.
  • Matrix Entry a₂₂ (Bottom-Right): Row 2, Column 2.

What the calculation includes

With the default values shown, Matrix Determinant (det A) & Binary/Hex Radix Representation is det(A) = 14 • 0xFF (11111111₂). The calculation follows 2×2 Matrix Cramer Inversion & Positional Radix Expansion; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Matrix Entry a₁₁ (Top-Left) from 4 to 5 changes Matrix Determinant (det A) & Binary/Hex Radix Representation from det(A) = 14 • 0xFF (11111111₂) to det(A) = 19 • 0xFF (11111111₂). This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses standard IEEE/positional base representations.
  • Singular matrices (det = 0) do not have an inverse.

Scope: Global Check the assumptions and limitations against your situation before relying on the estimate.

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