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Matrix Calculator

2x2 Determinant: |a b; c d| = ad - bc

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How this matrix calculator works

Enter the four entries of a 2×2 matrix: a and b on the first row, c and d on the second. The calculator finds the determinant with Determinant = a×d − b×c. For a=1, b=2, c=3, d=4 that is 1×4 − 2×3 = −2.0000. The result is shown to four decimal places.

Layout matches the on-page formula |a b; c d|. a is top-left, b top-right, c bottom-left, and d bottom-right. The identity matrix 1, 0, 0, 1 gives 1.0000. A diagonal matrix 2, 0, 0, 2 gives 4.0000. All four fields must be filled with finite numbers; otherwise the result stays “--.”

A non-zero determinant means the matrix is invertible. A zero determinant means it is singular: the rows are multiples of each other. a=2, b=3, c=4, d=6 gives 0.0000 because the second row is twice the first. Negatives are normal: a=5, b=3, c=2, d=1 gives −1.0000.

This page only computes the 2×2 determinant. It does not multiply matrices, find an inverse, or handle larger sizes. Check by hand with a×d minus b×c: for a=3, b=1, c=2, d=4 you get 12 − 2 = 10.0000.

2×2 determinant examples

Each row is a×d − b×c, rounded to four decimals as on the calculator.

abcdDeterminantWhat it marks
1234−2.00001×4 − 2×3
312410.00003×4 − 1×2
10011.0000Identity matrix
20024.0000Scaled identity
5321−1.0000Negative determinant
23460.0000Singular matrix
0110−1.00000×0 − 1×1
1.52340.0000Decimal entries

How to read the result

  • ad − bc

    1×4 − 2×3 = −2.0000.

  • Non-zero

    The 2×2 matrix is invertible.

  • Zero

    Singular matrix; rows are dependent.

  • Four decimals

    10 shows as 10.0000.

Frequently asked questions

  • What does this matrix calculator find?

    The determinant of a 2×2 matrix. Enter a, b, c, and d for the matrix |a b; c d|. The formula is Determinant = a×d − b×c. For a=1, b=2, c=3, d=4 the result is 1×4 − 2×3 = −2.0000, shown to four decimal places.

  • How are a, b, c, and d arranged?

    a is top-left, b is top-right, c is bottom-left, and d is bottom-right. The first row is a b. The second row is c d. That matches the on-page note |a b; c d| = ad − bc.

  • Why does the determinant matter?

    For a 2×2 matrix it is a single number that tells you whether the matrix is invertible. If the determinant is not zero, the matrix has an inverse. If it is zero, the matrix is singular and has no inverse. a=2, b=3, c=4, d=6 gives 0.0000.

  • What is the identity matrix result?

    The 2×2 identity is a=1, b=0, c=0, d=1. Then 1×1 − 0×0 = 1.0000. A scaled version such as a=2, b=0, c=0, d=2 gives 4.0000.

  • Can the determinant be negative?

    Yes. a=1, b=2, c=3, d=4 gives −2.0000. a=5, b=3, c=2, d=1 gives 5×1 − 3×2 = −1.0000. The sign is part of the value.

  • What inputs are required?

    All four boxes must hold finite numbers. An empty field or a non-finite value leaves the result as “--.” Decimals are allowed, such as a=1.5, b=2, c=3, d=4.

  • Does this multiply or invert matrices?

    No. This page only computes the 2×2 determinant ad − bc. It does not multiply two matrices, find an inverse, or handle 3×3 matrices.

  • How precise is the answer?

    The display uses four decimal places. 3×4 − 1×2 = 10 becomes 10.0000. A zero determinant shows 0.0000.

  • How can I check the answer by hand?

    Multiply a by d, multiply b by c, then subtract the second product from the first. For a=3, b=1, c=2, d=4: 3×4 = 12 and 1×2 = 2, so 12 − 2 = 10.0000.

  • What does a zero determinant mean for the rows?

    The two rows are linearly dependent: one row is a multiple of the other. In a=2, b=3, c=4, d=6, the second row is twice the first, so ad − bc = 0.