Problem 53 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the matrix equation is \(X=A^{-1}B\), where \(A^{-1}\) is the inverse of matrix \(A\)

Step 1: Confirm Matrix A is Invertible

Examine matrix \(A\) to confirm that it is invertible. This is crucial as the inverse of a singular matrix (i.e., a matrix that is not invertible) does not exist.

Step 2: Calculate the Inverse of Matrix A

If matrix \(A\) is invertible, calculate its inverse. The inverse of a matrix \(A\) is often denoted by \(A^{-1}\). Use a suitable method such as the Gaussian elimination, adjugate, or Laplace expansion method to calculate the inverse.

Step 3: Apply the Inverse to Both Sides of the Equation

Multiply the inverse of matrix \(A\), that is, \(A^{-1}\), to both sides of the equation. The equation now reads as \(A^{-1}AX=A^{-1}B\)

Step 4: Simplify the Equation

Simplify the equation. Notably, the product of a matrix and its inverse yields the identity matrix. The left-hand side of the equation, \(A^{-1}A\), simplifies to the identity matrix, denoted by \(I\). Thus, \(IX=A^{-1}B\)

Step 5: Isolate Matrix X

Finally, multiplying the identity matrix by matrix \(X\) simply yields matrix \(X\) (since \(I\) times any matrix equals the original matrix). Thus, \(X=A^{-1}B\), which is the solution to the matrix equation.