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To determine for which values of \( c \) the matrix \( A = \begin{bmatrix} 1 & 1 \\ c & c^2 \end{bmatrix} \) is invertible, we need to check when the determinant of the matrix is non-zero. A matrix is invertible if and only if its determinant is not equal to zero.

### Step-by-step Solution:

1. **Find the Determinant of Matrix \( A \):**

   The determinant of a \( 2 \times 2 \) matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

   \[
   \text{det}(A) = ad - bc
   \]

   For matrix \( A = \begin{bmatrix} 1 & 1 \\ c & c^2 \end{bmatrix} \):

   \[
   \text{det}(A) = (1)(c^2) - (1)(c) = c^2 - c
   \]

2. **Determine When the Determinant is Zero:**

   Set the determinant equal to zero and solve for \( c \):

   \[
   c^2 - c = 0
   \]

   Factor the quadratic equation:

   \[
   c(c - 1) = 0
   \]

   This equation has two solutions:

   \[
   c = 0 \quad \text{or} \quad c = 1
   \]

3. **Conclusion:**

   The matrix \( A \) is invertible for all values of \( c \) except when \( c = 0 \) or \( c = 1 \).

### Answer:

- For all \( c \) such that \( c \neq 0 \) and \( c \neq 1 \), the matrix \( A \) is invertible.

If you have any questions or need further details, feel free to ask!

### Relative Questions:

1. How do you find the inverse of a \( 2 \times 2 \) matrix if it is invertible?
2. What is the geometric interpretation of a matrix being non-invertible?
3. How does the determinant of a matrix relate to its eigenvalues?
4. What is the significance of a matrix's determinant being zero?
5. How can you generalize this approach to larger matrices?

### Tip:
Always check for zero determinants when determining the invertibility of a matrix.

For what values of c will the matrix A = [[1, 1], [c, c^2]] be invertible?

Discover for which values of c the matrix A = [[1, 1], [c, c^2]] is invertible. Learn key linear algebra concepts like matrix determinants and the conditions for matrix inversion, suitable for Grades 10-12.

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