Math & Algebra

Free Matrix Determinant & Transpose Calculator — 2x2 & 3x3

Determinants are fundamental scalar values associated with square matrices that reveal whether a linear system has a unique solution and whether a matrix is invertible. If $det(A) eq 0$, the matrix is non-singular and invertible. Our free matrix determinant calculator helps linear algebra students evaluate inverse matrix calculator prerequisites, solve 3x3 matrix determinant solver matrices, and compute matrix transpose calculator outputs.

Matrix Determinant & Transpose Calculator

Calculate determinants $\det(A)$, transpose $A^T$, and check invertibility for 2x2 and 3x3 matrices.

Determinant det(A) or |A|

-42

Matrix A is INVERTIBLE (det ≠ 0).

Dimensions:3x3 Square Matrix

Invertible Status:Non-Singular

Matrix Transpose (Aᵀ) Matrix

Mathematical Formulas & Calculation Methods

Matrix Determinant Formulas

2x2 and 3x3 cofactor expansion determinant equations.

2x2: det(A) = ad - bc 3x3: det(A) = a11(a22 a33 - a23 a32) - a12(a21 a33 - a23 a31) + a13(a21 a32 - a22 a31)

Worked Step-by-Step Examples

Example 1: 3x3 Matrix [[2, 3, 1], [4, 1, 5], [0, 2, 3]]

"Calculate determinant det(A)."

Steps:

Term 1: 2 × (1*3 - 5*2) = 2 × (3 - 10) = -14.
Term 2: -3 × (4*3 - 5*0) = -3 × 12 = -36.
Term 3: +1 × (4*2 - 1*0) = +1 × 8 = +8.
det(A) = -14 - 36 + 8 = -42.

Result: det(A) = -42 (Matrix is Invertible!).

What is Matrix Determinant & Transpose Calculator?

A matrix determinant calculator evaluates linear system independence and transformation scaling factor for square matrices.

The transpose of a matrix (Aᵀ) swaps rows and columns such that entry (i, j) becomes entry (j, i).

How to Use Matrix Determinant & Transpose Calculator

1Select matrix dimension (2x2 or 3x3).
2Enter matrix element coefficients.
3Review instant determinant det(A), invertibility status, and Transpose matrix Aᵀ.

Key Features

Supports 2x2 and 3x3 square matrix sizes.
Displays Matrix Transpose Aᵀ matrix.
Tests for non-singular (det ≠ 0) vs singular matrix condition.

Benefits of Using Matrix Determinant & Transpose Calculator

Essential tool for linear algebra, physics, and computer graphics students.

Eliminates manual sign and cofactor multiplication mistakes.

Practical Use Cases

Linear Algebra Students

Solve matrix determinants, Cramer's rule, and inverse conditions.

Engineers & Programmers

Verify 3D graphic transformation matrices and determinants.

Frequently Asked Questions

What does a zero determinant mean?

A determinant of zero (det = 0) means the matrix is singular and has no inverse, indicating its rows/columns are linearly dependent.

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