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How is an inverse matrix used to solve the equation AX = B?

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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.

To solve the matrix equation $A X = B$ , where $A$ and $B$ are known matrices and $X$ is an unknown matrix, you can use the inverse matrix $A^{- 1}$ , provided that $A$ is a nonsingular matrix (meaning its determinant is not zero).12

The process involves these steps:

Left-multiplication: Since $A^{- 1}$ exists, you multiply both sides of the equation on the left by $A^{- 1}$ :
$A^{- 1} (A X) = A^{- 1} B$ 1
Associativity: Using the associative property of matrix multiplication, the left side becomes:
$(A^{- 1} A) X = A^{- 1} B$ 1
Identity: Because $A^{- 1} A = I$ (where $I$ is the identity matrix), the equation simplifies to:
$I X = A^{- 1} B$ 1
Solution: Since $I X = X$ , the unique solution is:
$X = A^{- 1} B$ 1

This method provides a direct algebraic approach to finding the unknown matrix $X$ . While this is a fundamental theoretical application, systems of linear equations are also frequently solved using Gauss elimination, a process that is highly efficient for computational implementation.13