Question

Arrange the following in correct order to solve a linear system \(AX=T\) using matrix inversion. A. Verify that determinant of \(A\) is non-zero B. Compute the adjoint C. Compute \(A^{-1}\) D. Multiply \(A^{-1}T\) to get \(X\) E. Set up the co-efficient Matrix \(A\) and constant Matrix \(T\).

• A. E, A, B, C, D
• B. A, B, C, D, E
• C. E, B, A, D, C
• D. E, A, B, D, C

Hint:

To solve \(AX=T\) by matrix inversion, first form \(A\) and \(T\), check \(|A|\neq0\), find adjoint, find inverse, then calculate \(X=A^{-1}T\).

Answer 1

The Correct Option is A

Concept: A system of linear equations can be solved by matrix inversion if the coefficient matrix is invertible. \[ AX=T \] If \(A^{-1}\) exists, then: \[ X=A^{-1}T \]

Step 1: Set up coefficient matrix and constant matrix.
First, write the system in matrix form by forming the coefficient matrix \(A\) and constant matrix \(T\). \[ AX=T \] So the first step is E.

Step 2: Check whether inverse exists.
The inverse of \(A\) exists only when determinant of \(A\) is non-zero. \[ |A|\neq 0 \] So the second step is A.

Step 3: Compute the adjoint.
To find inverse using the adjoint method, compute the adjoint of matrix \(A\). \[ \text{adj}(A) \] So the third step is B.

Step 4: Compute the inverse matrix.
The inverse is calculated as: \[ A^{-1}=\frac{\text{adj}(A)}{|A|} \] So the fourth step is C.

Step 5: Find the solution matrix.
Finally, multiply both sides by \(A^{-1}\): \[ X=A^{-1}T \] So the fifth step is D. Therefore, the correct order is: \[ E,\ A,\ B,\ C,\ D \]