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Calculator Answers

Determinant Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A determinant calculator is a linear algebra tool used to evaluate the determinant of square matrices, specifically of 2x2 and 3x3 sizes. It multiplies diagonal elements and subtracts anti-diagonal products, expanding larger matrices by using minor sub-matrices recursively. The determinant is used to determine matrix invertibility, solve linear equation systems using Cramer's rule, and calculate scaling areas or volumes.

Calculate the determinant of a 2x2 or 3x3 matrix instantly with step-by-step minor expansions.

Quick Answer

Compute the determinant of a 2x2 or 3x3 matrix. Enter your matrix values to see the calculated scalar result and expansion steps.

Matrix Dimensions

Matrix Entries

Determinant Result

Determinant of 2x2 Matrix

-2

Calculation Processdet(A) = (1 × 4) − (2 × 3) = 4 − 6 = -2

The determinant characterizes whether the system of linear equations represented by this matrix has a unique solution.

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Examples

2x2 Matrix [[3, 8], [4, 6]]

det = (3 × 6) − (8 × 4) = 18 − 32 = -14

3x3 Matrix [[1, 2, 3], [0, 4, 5], [1, 0, 6]]

det = 1(24) − 2(-5) + 3(-4) = 24 + 10 − 12 = 22

How it works

The calculator performs cross-multiplication of elements along the diagonals and subtracts the product of the anti-diagonal elements, scaling up recursively for 3x3 matrices.

2x2 Formula · ad − bc

3x3 Formula · a11(M11) − a12(M12) + a13(M13)

How to calculate the determinant

To find the determinant of a square matrix, apply expansion by minors. This method breaks down larger matrices into smaller 2x2 sub-matrices.

For a 2x2 matrix:

det([[a, b], [c, d]]) = ad − bc

For a 3x3 matrix:

det([[a, b, c], [d, e, f], [g, h, i]]) = a(ei − fh) − b(di − fg) + c(dh − eg)

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Matrix CalculatorRun matrix operations on 2x2 and 3x3 matrices: addition, subtraction, scalar multiplication, matrix multiplication, transpose, determinant, and inverse. Not a system solver (see RREF and system of equations).

Frequently asked questions

The determinant is a scalar value calculated from a square matrix. It provides important properties of the matrix, such as whether it has an inverse (determinant is non-zero) and how the matrix scales areas or volumes in linear transformations.

For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated using the formula: det(A) = ad − bc. You multiply the elements of the main diagonal and subtract the product of the elements of the other diagonal.

For a 3x3 matrix, you expand it along the first row: det(A) = a11 × (a22×a33 − a23×a32) − a12 × (a21×a33 − a23×a31) + a13 × (a21×a32 − a22×a31). This reduces the 3x3 determinant to three 2x2 determinants.

A determinant of zero means the matrix is 'singular' or non-invertible. In terms of transformations, it means the matrix collapses space into a lower dimension (e.g., squashing a 2D plane into a 1D line), so its area scale factor is 0.

Yes. A negative determinant indicates that the transformation changes the orientation of space (like a reflection or mirror image) in addition to scaling it.