2x2 Determinant Calculator

This calculator finds the determinant of a 2 by 2 matrix, one of the first calculations you meet when working with matrices in algebra, statistics or engineering. You enter the four entries of the matrix: a and b from the top row, and c and d from the bottom row. The calculator instantly returns the determinant, calculated as a times d minus b times c, and displays it alongside two extra results, whether the matrix is invertible (yes if the determinant is non-zero, no if it is singular) and the matrix laid out so you can double check what you typed. The determinant is more than a number to memorise; it tells you whether a matrix can be inverted, it measures how the matrix scales area when used as a transformation, and it underpins Cramer's rule for solving simultaneous equations and the vector cross product. A worked example is included below, using the matrix with rows (1, 2) and (3, 4), where the determinant works out to 1 times 4 minus 2 times 3, which equals minus 2. Use this tool to check homework, verify a hand calculation, or quickly test whether a matrix is singular before you try to invert it. Just make sure you enter each value in the right position, since swapping a, b, c or d around will change the result.

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a (row1)

b (row1)

c (row2)

d (row2)

Invertible?

Matrix

The formula

For the matrix with rows (a, b) and (c, d), the determinant is ad minus bc. The matrix is invertible exactly when this is non-zero, and the determinant equals the signed area of the parallelogram spanned by its rows.

Worked example

For the matrix with rows (1, 2) and (3, 4), the determinant is 1 times 4 minus 2 times 3 = 4 minus 6 = minus 2. Enter 1, 2, 3, 4 to confirm.

Frequently asked questions

How do I find a 2x2 determinant?

Multiply the main diagonal (top-left times bottom-right) and subtract the other diagonal (top-right times bottom-left): ad minus bc.

What does the determinant tell me?

Whether the matrix is invertible (non-zero) and how it scales area. Zero means the rows are parallel and the matrix is singular.

Where is it used?

In matrix inverses, Cramer’s rule, area and the vector cross product.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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