Matrix Calculator (2×2 Add, Multiply, Determinant & Inverse)

Use this free matrix calculator to work with 2×2 matrices. Add or subtract two matrices, multiply them, take a transpose, multiply by a scalar, or find the determinant and inverse of a matrix — each result is shown with the element-by-element computation.

How to use the matrix calculator

1. Enter the numbers for matrix A (and matrix B when the operation needs it).
2. Choose an operation: add, subtract, multiply, determinant, inverse, transpose or scalar.
3. Read the resulting matrix and the worked steps.

Key formulas

The determinant of a 2×2 matrix [[a, b], [c, d]] is ad − bc. The inverse is 1/(ad − bc) times [[d, −b], [−c, a]] — which only exists when the determinant is not zero. Matrix multiplication combines rows of A with columns of B.

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Frequently asked questions

How do you find the determinant of a 2×2 matrix?

For [[a, b], [c, d]], the determinant is ad − bc. For [[1, 2], [3, 4]] that is (1)(4) − (2)(3) = −2.

When does a matrix have no inverse?

When its determinant is zero. Such a matrix is called singular, and the calculator tells you no inverse exists.

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How does matrix multiplication work?

Each entry of the product is the dot product of a row of A with a column of B. The calculator shows each entry’s a·e + b·g computation.

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Read the full lesson: learn the method step by step.

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