22.4.2. Method 2: Using Minors

Try these first

1. 1.

   What is the rank of a matrix if all rows are linearly independent?

   Hint

   Think about linear independence!

2. 2.

   Define a minor in the context of matrices.

   Hint

   Consider the relationship with determinants.

3. 3.

   What method are we using to determine the rank of the matrix?

   • Using minors
   • Gaussian elimination
   • Echelon forms

   Hint

   Think about the method that involves determinants!

4. 4.

   True or False: The rank of a matrix can be determined solely by the size of the matrix.

   • True
   • False

   Hint

   Consider what affects linear independence!

5. 5.

   Consider the matrix $C = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 2 & 4 & 6 \end{bmatrix}$. Determine its rank using minors.

   Hint

   Reflect on linear dependencies!

6. 6.

   For the matrix $D = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 0 & 4 \\ 3 & 6 & 7 \end{bmatrix}$, compute the rank and explain.

   Hint

   Examine the submatrices carefully for dependencies.