Math Problem Statement

Solve the problem

Let F = [[a, b], [c, d]] be an invertible matrix. Prove or disprove the following statements: (a) If you multiply F by a scalar k, then (kF)^{-1} = k^{-1}F^{-1}. (b) If you add two invertible matrices F1 + F2, then (F1 + F2)^{-1} = F1^{-1} + F2^{-1}. (c) If you transpose a matrix, then (F^T)^{-1} = (F^{-1})^T.

Solution

Let’s analyze and solve the problem step by step. For each subproblem, we will either prove or disprove the statement.

Problem 3(a):

If FF is an invertible matrix and kk is a scalar, prove or disprove:
(kF)−1=k−1F−1.(kF)^{-1} = k^{-1} F^{-1}.

Solution:

The inverse of a scalar multiple of a matrix kFkF is given by the formula: (kF)−1=1kF−1.(kF)^{-1} = \frac{1}{k} F^{-1}. This formula is derived as follows:

  1. By definition, (kF)(kF)−1=I(kF)(kF)^{-1} = I, where II is the identity matrix.
  2. Multiply both sides by 1k\frac{1}{k}:
    1k(kF)(kF)−1=1kI.\frac{1}{k}(kF)(kF)^{-1} = \frac{1}{k} I.
  3. Simplify: F(kF)−1=1kI  ⟹  (kF)−1=1kF−1.F(kF)^{-1} = \frac{1}{k} I \implies (kF)^{-1} = \frac{1}{k} F^{-1}.

Thus, the statement is true.

Problem 3(b):

If F1F_1 and F2F_2 are invertible matrices, prove or disprove: (F1+F2)−1=F1−1+F2−1.(F_1 + F_2)^{-1} = F_1^{-1} + F_2^{-1}.

Solution:

This statement is generally false. Here’s a counterexample to disprove it:

Let:

F_2 = \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}.$$ Then: $$F_1 + F_2 = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}.$$ The matrix $$F_1 + F_2$$ is not invertible (its determinant is 0). However, both $$F_1$$ and $$F_2$$ are invertible. This contradiction disproves the statement. --- ### **Problem 3(c):** If $$F$$ is an invertible matrix, prove or disprove: $$(F^T)^{-1} = (F^{-1})^T.$$ #### **Solution:** This statement is **true**. The proof is as follows: 1. Start with the property of transposes: $$(AB)^T = B^T A^T.$$ 2. Let $$F$$ be invertible, so $$F F^{-1} = I$$, where $$I$$ is the identity matrix. 3. Transpose both sides: $$(F F^{-1})^T = I^T.$$ 4. Apply the property of transposes: $$(F^{-1})^T F^T = I.$$ 5. By the definition of inverses, this implies: $$(F^T)^{-1} = (F^{-1})^T.$$ Thus, the statement is **true**. --- ### **Summary of Results:** - (a) True: $$(kF)^{-1} = k^{-1}F^{-1}$$. - (b) False: $$(F_1 + F_2)^{-1} \neq F_1^{-1} + F_2^{-1}$$. - (c) True: $$(F^T)^{-1} = (F^{-1})^T$$. Let me know if you have questions or need further explanation! ### **5 Related Questions:** 1. What conditions must $$F_1$$ and $$F_2$$ satisfy for $$(F_1 + F_2)^{-1} = F_1^{-1} + F_2^{-1}$$ to hold true? 2. How do you compute the inverse of a block matrix? 3. Can you generalize the property of scalar multiples to non-square matrices? 4. What is the significance of the transpose in inverse operations? 5. How does the determinant of $$F$$ relate to its inverse? ### **Tip:** For any matrix operations involving inverses, always check the invertibility of the matrices involved—this ensures the operations are valid!

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Math Problem Analysis

Mathematical Concepts

Matrix operations
Matrix inversion
Matrix transpose
Linear algebra

Formulas

(kF)^{-1} = k^{-1}F^{-1}
(F^T)^{-1} = (F^{-1})^T
Counterexamples to matrix inversion properties

Theorems

Inverse matrix properties
Transpose of a product of matrices
Definition of invertibility

Suitable Grade Level

Undergraduate Mathematics or Linear Algebra Course

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