Problem 61 - Algebra and Trigonometry

Question

Explain how to solve the matrix equation \(A X=B\)

Step-by-Step Solution

Answer

The solution of the matrix equation A*X = B is X = A-inverse * B if A is invertible (i.e., A is square and its determinant is not zero). First calculate the inverse of A and then multiply it with B to get X.

Step 1: Understanding the Concept

Matrix X is the solution of the matrix equation A*X = B if A*X is equal to B for that particular X. For matrix multiplication A*X to be defined, the number of columns in A needs to be equal to the number of rows in X. Moreover, for A’s inverse to exist, A needs to be a square matrix (i.e., have the same number of rows and columns) and be invertible or non-singular (its determinant is not equal to zero). If these conditions are met, we can find X by multiplying the inverse of A with B.

Step 2: Calculate Inverse of A

First of all, you have to determine whether the inverse of A exists or not by checking if A is a square matrix and its determinant is non-zero. If it does, then calculate the inverse of A. You can use several methods to calculate the inverse of a matrix including the adjugate method, Gaussian elimination or using a formula if A is a 2x2 matrix.

Step 3: Multiply A-inverse with B

Multiply the inverse of A with B to get X. When you multiply a matrix by its inverse the result is the identity matrix. So when you multiply the inverse of A with both sides of the original equation A*X = B, the left hand side simplifies to X and hence X equals A-inverse multiplied by B. Be careful with matrix multiplication as it is not commutative, meaning that A*B is not always equal to B*A.

Step 4: Conclusion

After performing the above steps, we obtain the values for X that satisfy the given matrix equation A*X = B. However, if the inverse of A does not exist then the given equation does not have a unique solution.