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How does an inverse matrix help solve a system of linear equations?

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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).

An inverse matrix provides a direct method for solving a system of linear equations expressed in the matrix form $A X = B$ , where $A$ is the coefficient matrix, $X$ is the column vector of unknowns, and $B$ is the column vector of constants.1

If the matrix $A$ is nonsingular—meaning its determinant is non-zero—it possesses an inverse matrix, denoted as $A^{- 1}$ , such that $A A^{- 1} = A^{- 1} A = I$ , where $I$ is the identity matrix.12

To solve for the unknowns, you can multiply both sides of the equation $A X = B$ on the left by $A^{- 1}$ :

$A^{- 1} (A X) = A^{- 1} B$

Using the associative property of matrix multiplication, this simplifies to:

$(A^{- 1} A) X = A^{- 1} B$
$I X = A^{- 1} B$
$X = A^{- 1} B$

By calculating the product of the inverse matrix and the constant vector, you isolate the unknown variables, providing a unique solution to the system. This approach is one of several ways to handle systems of equations, alongside methods like Gauss elimination.13