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Medium

The Determinant of a 2×2 Matrix

Simple Explanation

The determinant of a 2×2 matrix A = [[a,b],[c,d]] is the single number ad − bc, written det(A) or |A|. A matrix has an inverse exactly when its determinant is nonzero.

Why Do We Need It?

The determinant is a fast diagnostic check for whether a matrix can be inverted at all, and it is the key ingredient in the inverse formula itself.

Formula

Determinant of a 2×2 Matrix

For A = [[a, b], [c, d]]: det(A) = ad − bc

A single number computed from a 2×2 matrix that reveals whether the matrix has an inverse — a matrix has an inverse exactly when its determinant is nonzero.

a, b, c, d
— the four entries of the 2×2 matrix, read left-to-right, top-to-bottom
det(A)
— the determinant of A, also written |A|

When to use it: Whenever you need to check if a 2×2 matrix has an inverse, or as the first step of computing that inverse.

Worked Example

Evaluate a determinant

Find the determinant of A = [[3, 4], [2, 5]].

    Why Does This Work?

    The determinant measures how a matrix scales area when used as a geometric transformation — when det(A)=0, the transformation squashes every shape down to a line or a point (losing a dimension), which is exactly the situation where the transformation cannot be undone (no inverse exists).

    Real-Life Example

    Checking whether a system of equations has a unique solution

    Before attempting to solve a system of two linear equations using matrices, it helps to know in advance whether a unique solution even exists.

    Computing the determinant of the coefficient matrix instantly reveals this — nonzero means a unique solution exists; zero means it does not (no solution, or infinitely many).

    Practice

    Find the determinant of [[6, 2], [5, 3]].

    Medium

    Common mistake

    Computing bc − ad instead of ad − bc — the order matters; always multiply the main diagonal (top-left × bottom-right) first, then subtract the off-diagonal product.

    Quick Review

    • det(A) = ad − bc, for A = [[a,b],[c,d]].
    • A matrix has an inverse exactly when det(A) ≠ 0.
    • The first step in finding a 2×2 matrix's inverse.
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