This problem set is extensive and involves various operations on matrices, including matrix addition, multiplication, determinants, adjugates, inverses, and finding specific values to make matrices invertible. Let's break down the solution approach for each exercise and provide guidance on how to solve each part.

Exercise 1:

Given Matrices:

You have matrices $A_1, A_2, A_3, A_4, A_5,$ and $A_6$ defined explicitly.

Tasks:

1. Matrix Operations:

   • $A_1 + A_2$, $A_1 + A_3$, $A_1^T$, $3A_1$, $-2A_3$, $A_1 \cdot A_2$, etc.
   • These involve basic matrix addition, scalar multiplication, and matrix multiplication.
   • To compute these, add or multiply the elements of the matrices as specified.

2. Finding Determinants and Adjugates:

   • Determinants $\det(A_1), \det(A_2), \ldots$: Use the formula for $3 \times 3$ determinants.
   • Adjugates $\text{adj}(A_1), \text{adj}(A_2), \ldots$: The adjugate is the transpose of the cofactor matrix. This involves finding cofactors for each element and arranging them appropriately.

3. Inverses $A_1^{-1}, A_2^{-1}, \ldots$:

   • If $\det(A) \neq 0$, the inverse exists and can be found by $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$.
   • Check the determinants calculated earlier to ensure they are non-zero before finding the inverse.

4. Ranks $\text{rank}(A_1), \text{rank}(A_2), \ldots$:

   • Use row-reduction to determine the rank of each matrix.

5. Solving Matrix Equations:

   • For $A X = A_2$, solve for $X$ by finding $A^{-1}$ if it exists and multiplying both sides by it.
   • Similar approach for other matrix equations.

Exercise 2:

Finding $\lambda$ for Invertibility:

1. Condition for Invertibility:
   • A matrix is invertible if its determinant is non-zero.
2. Determinant Calculation:
   • Set up the determinant for the given matrix with $\lambda$ and solve for $\lambda$ such that $\det = 0$.
3. Solution for $\lambda$:
   • Solve the resulting equation for $\lambda$ to determine the value(s) that make the matrix invertible.

Would you like detailed calculations for each of these steps, or is there a specific part of the exercise you’d like help with first?

Here are some follow-up questions that might help guide further understanding:

1. Would you like help calculating the determinant of a specific matrix?
2. Do you need assistance with matrix multiplication or transposing steps?
3. Are there any matrix rank calculations you’re unsure about?
4. Do you need help with solving for $\lambda$ in Exercise 2?
5. Would you like guidance on calculating the adjugate for a matrix?

Tip: Remember that the rank of a matrix is the maximum number of linearly independent row or column vectors.