True or False: Given the 𝐴 is an invertible matrix satisfying the equation 𝐴𝑋 equals 𝐵, the solution of this matrix equation is 𝑋 equals 𝐴 inverse 𝐵.
A matrix equation is an equation where the variables stand for matrices. We’ve been asked if the solution of the matrix equation 𝐴𝑋 equals 𝐵 is 𝑋 equals 𝐴 inverse 𝐵. In order to find a solution to this matrix equation, there’s one important thing to recall, that is, that 𝐴 inverse multiplied by 𝐴 equals the identity matrix, and that’s the same as 𝐴 multiplied by 𝐴 inverse. So if we take 𝐴𝑋 equals 𝐵 and we multiply on the left by 𝐴 inverse, we’ve got 𝐴 inverse multiplied by 𝐴 multiplied by 𝑋 equals 𝐴 inverse multiplied by 𝐵. This is okay only because we’re told that 𝐴 is an invertible matrix.
Now remember that matrix multiplication is associative. This means that this is the same as 𝐴 inverse multiplied by 𝐴 multiplied by 𝑋 equals 𝐴 inverse multiplied by 𝐵. Now by definition, we know that 𝐴 inverse multiplied by 𝐴 gives us the identity matrix. But we know that taking a matrix and multiplying it by the identity matrix just gives us that original matrix, so we have that 𝑋 equals 𝐴 inverse 𝐵. And therefore, this statement is true. This is a particularly useful result because we can use this to help us solve systems of linear equations.