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How does a nonzero determinant show that a square matrix has an inverse?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, a square matrix is considered invertible—also known as nonsingular—if and only if its determinant is nonzero.1

The determinant serves as a critical indicator of a matrix's properties. When the determinant is nonzero, it ensures that the matrix does not collapse the space it acts upon, meaning the transformation it represents is reversible. Conversely, if the determinant is zero, the matrix is singular, indicating that it lacks a multiplicative inverse.12

This relationship is fundamental because the process of matrix inversion requires the ability to perform operations that are impossible if the determinant is zero, such as those involving a full row or column of zeros. Furthermore, a nonzero determinant guarantees that a system of $n$ equations in $n$ unknowns, represented by $A X = B$ , will always have a unique solution.12