Solving Linear Systems with Matrix Inverses (Exploration: Exploring Matrix Inverses)
Solve systems of linear equations in Grade 10 using matrix inverses: express AX = B, compute A⁻¹, then multiply both sides to find the solution matrix X.
Key Concepts
New Concept Two matrices $A$ and $B$ are inverses of each other if $A \cdot B = B \cdot A = I$.
What’s next Next, you'll calculate the inverse of a matrix and use it as a powerful new tool to solve systems of linear equations.
Common Questions
How do you set up a matrix equation to solve a linear system?
Write the system as AX = B, where A is the coefficient matrix, X is the variable column matrix, and B is the constant column matrix.
How do you solve AX = B using the matrix inverse?
Multiply both sides on the left by A⁻¹: X = A⁻¹B. This works only if the matrix A is invertible (det(A) ≠ 0).
When does the matrix inverse method fail?
It fails when the coefficient matrix A has a determinant of zero (singular matrix). This indicates the system has no unique solution — it is either inconsistent or dependent.