### Video Transcript

True or False: Given the π΄ is an invertible matrix satisfying the equation π΄π equals π΅, the solution of this matrix equation is π equals π΄ inverse π΅.

A matrix equation is an equation where the variables stand for matrices. Weβve been asked if the solution of the matrix equation π΄π equals π΅ is π equals π΄ inverse π΅. In order to find a solution to this matrix equation, thereβs one important thing to recall, that is, that π΄ inverse multiplied by π΄ equals the identity matrix, and thatβs the same as π΄ multiplied by π΄ inverse. So if we take π΄π equals π΅ and we multiply on the left by π΄ inverse, weβve got π΄ inverse multiplied by π΄ multiplied by π equals π΄ inverse multiplied by π΅. This is okay only because weβre told that π΄ is an invertible matrix.

Now remember that matrix multiplication is associative. This means that this is the same as π΄ inverse multiplied by π΄ multiplied by π equals π΄ inverse multiplied by π΅. Now by definition, we know that π΄ inverse multiplied by π΄ 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 π equals π΄ inverse π΅. 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.