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How does a determinant help determine whether a system of two linear equations has a unique solution?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In linear algebra, the determinant of the coefficient matrix is a critical value used to determine if a system of linear equations has a unique solution.

For a system of two linear equations with two unknowns, represented by the matrix equation $A X = B$ , the determinant ( $Δ$ ) is calculated as $a_{11} a_{22} - a_{12} a_{21}$ . The relationship between the determinant and the system's solution is as follows:123

Unique Solution: If the determinant is nonzero ( $Δ \neq 0$ ), the matrix $A$ is nonsingular (invertible), which guarantees that the system has a unique solution.124
No Unique Solution: If the determinant is zero ( $Δ = 0$ ), the matrix is singular, and the system does not have a unique solution. In such cases, the system may have no solution at all or infinitely many solutions, depending on the constants involved.54

This principle is foundational to Cramer’s rule, which utilizes determinants to express the values of the unknowns directly.53