Detailed Methodology
The Matrix Inverse Method (also known as the Inversion Method) is a powerful algebraic tool used to solve systems of linear equations. While students often learn this for 2x2 matrices in 9th grade, the 11th-grade curriculum (National Book Foundation) expands this to 3x3 systems, requiring a deep understanding of determinants, cofactors, and adjoints.
1. Matrix Formation
First, convert the system of linear equations into a matrix equation AX = B.
- Matrix A: Contains the coefficients of variables (x, y, z).
- Matrix X: The column matrix of variables [x, y, z].
- Matrix B: The constants on the right side of the equations.
2. The Determinant Check
Calculate |A| by expanding along the first row (R1).
Important:
If |A| = 0, the matrix is singular, and the system has no solution. Do not proceed further.
Finding the Adjoint of a 3x3 Matrix
This is the most time-consuming part. Follow the Co-factor Method meticulously:
Find the determinant of the 2x2 matrix remaining after hiding the row and column of an element.
Use the sign pattern:
+ - +
- + -
+ - +
Swap the rows and columns of your co-factor matrix to finally get Adj(A).
3. Solving for X
Once you have the inverse matrix A-1 = Adj(A) / |A|, multiply it by Matrix B.
| Operation | Description |
|---|---|
| Scalar Division | Raja Hamza Khan recommends dividing by the determinant (|A|) at the very last step to avoid working with messy fractions too early. |
| Row × Column | Multiply the rows of A-1 by the single column of B. |
| Verification | Always plug your final values (x, y, z) back into the original equations to check for accuracy. |
Special Case: Question 6 Part 4
When variables appear in the denominator (e.g., 2/x + 3/y + 10/z = 4), Hamza explains a clever workaround:
Solve for [u, v, w] as usual, then find the reciprocal of your answers to get the final values of [x, y, z].