Let's go through the problem step by step:

(a) Prove or disprove: If $B = A^{-1}$, then $A = B^{-1}$.

Proof:

If $B = A^{-1}$, then we know that: $A B = A A^{-1} = I$ where $I$ is the identity matrix. This is the definition of the inverse matrix, meaning that $A$ and $B$ are inverses of each other. Therefore, it also follows that: $B A = B A^{-1} = I$ This implies that $A = B^{-1}$, since $B$ is the inverse of $A$, meaning the statement is true.

(b) Prove or disprove: The sum of two invertible matrices need not be an invertible matrix.

Disproof by example:

Let’s take two invertible matrices: $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ Both $A$ and $B$ are invertible, since their determinants are non-zero ($\det(A) = 1$ and $\det(B) = -1$). However, their sum is: $A + B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ The resulting matrix is the zero matrix, which is not invertible (since its determinant is zero). Therefore, this statement is true—the sum of two invertible matrices does not have to be invertible.

(c) Prove or disprove: The sum of two singular matrices of the same order cannot be invertible.

Proof:

A singular matrix is one that is not invertible, meaning its determinant is zero. However, it’s possible for the sum of two singular matrices to be invertible. Consider the following example: $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix}$ Both matrices $A$ and $B$ are singular since $\det(A) = 0$ and $\det(B) = 0$. Now, consider their sum: $A + B = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$ The determinant of $A + B$ is: $\det(A + B) = (0)(1) - (1)(1) = -1$ Since $\det(A + B) \neq 0$, $A + B$ is invertible. Therefore, the statement is false—the sum of two singular matrices can be invertible.

Let me know if you would like more details on any of the proofs or concepts. Here are 5 related questions for further exploration:

1. How do you compute the inverse of a $2 \times 2$ matrix?
2. Can the product of two singular matrices be invertible?
3. Under what conditions is the sum of two invertible matrices also invertible?
4. How does matrix addition affect the rank of the resulting matrix?
5. What properties of matrices ensure that their sum is singular?

Tip: When working with matrix inverses, always check if the matrix is invertible by calculating the determinant. If it's zero, the matrix is singular and non-invertible.