21.4.3. Methods to Find Inverse

Try these first

1. 1.

   Define what it means for a matrix to have an inverse.

   Hint

   Think about the identity matrix.

2. 2.

   What is the condition for a matrix to be non-singular?

   Hint

   Recall the property of determinants.

3. 3.

   What is the formula for the inverse using the adjoint method?

   • A⁻¹ = adj(A)/det(A)
   • A⁻¹ = det(A)/adj(A)
   • A⁻¹ = A*adj(A)

   Hint

   Recall the roles of adjugate and determinant.

4. 4.

   True or False: A singular matrix has an inverse.

   • True
   • False

   Hint

   Think about the determinant's role.

5. 5.

   Given the matrix A = $\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$, find the inverse using both methods: adjoint and Gauss-Jordan.

   Hint

   Calculate determinant first, for both methods.

6. 6.

   For a 4x4 matrix B, apply Gauss-Jordan elimination and find the inverse, showing each step explicitly. Consider B = $\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$.

   Hint

   Use row swaps and scaling as necessary.