Linear Algebra

Step 1: Understanding the Concept:
This problem uses the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation. For a 2x2 matrix, the characteristic equation is . The matrix will satisfy the equation .

Step 2: Key Formula or Approach:
1. Identify the trace ( ) and determinant ( ) from the given information.
2. Write the Cayley-Hamilton equation for matrix .
3. Manipulate this equation algebraically to find an expression for .

Step 3: Detailed Explanation:
We are given a 2x2 matrix with:


\end{itemize} According to the Cayley-Hamilton theorem for a 2x2 matrix: Substitute the given values into this equation: From this equation, we can express as: To find , we multiply the entire equation by : Now, we can substitute the expression for back into this equation:
Step 4: Final Answer:
The matrix is equal to .

Step 1: Understanding the Concept:
This question relates the concept of eigenvalues to matrix invertibility and determinants. A square matrix has an eigenvalue of zero if and only if it is singular, which means its determinant is zero.

Step 2: Key Formula or Approach:
1. The condition means that is the inverse of and is the inverse of . Both matrices are invertible.
2. An eigenvalue of a matrix is zero if and only if the determinant of the matrix is zero.
3. Use the property .

Step 3: Detailed Explanation:
We are given the relation between two square matrices and : Taking the determinant of both sides: We know that and the determinant of the identity matrix is 1. This equation implies that neither nor can be zero. If either were zero, their product would be zero, not one. A matrix has an eigenvalue of zero if and only if its determinant is zero. Since both and are non-zero, neither matrix nor matrix can have an eigenvalue of zero.

Step 4: Final Answer:
Therefore, zero is an eigenvalue of neither P nor Q.

Step 1: Understanding the Concept:
The given matrix is an augmented matrix representing a system of three linear equations in three variables ( ). We need to solve this system. We can convert the matrix back into equations and solve using substitution or elimination, or perform row operations on the matrix to get it into row-echelon form.

Step 2: Key Formula or Approach:
Convert the augmented matrix into a system of linear equations and solve. The augmented matrix corresponds to the system .

Step 3: Detailed Explanation:
The augmented matrix is: This translates to the following system of equations: \begin{align} x + y + z &= 3 &(1) \\ -2y - 2z &= 4 &(2) \\ x - 5y &= 5 &(3) \end{align} From equation (2), we can simplify by dividing by -2: From equation (3), we can express in terms of : Now, substitute equations (4) and (5) into equation (1): Simplify the equation to solve for : Now that we have , we can find and .
Substitute into equation (5): Substitute into equation (4):
Step 4: Final Answer:
The solution to the system of equations is , , and , which corresponds to option (C).

Step 1: Understanding the Concept:
The problem provides a polynomial equation that the matrix satisfies. According to the Cayley-Hamilton theorem, every square matrix satisfies its own characteristic equation. We can use the given equation to find an expression for the inverse of , , by manipulating the equation algebraically.

Step 2: Key Formula or Approach:
1. Start with the given matrix polynomial equation.
2. Isolate the term containing the identity matrix, .
3. Pre-multiply (or post-multiply) the entire equation by .
4. Simplify the resulting equation to solve for .

Step 3: Detailed Explanation:
We are given the equation: To find , we can isolate the term with the identity matrix : Since it is given that exists, we can pre-multiply both sides of the equation by : Using the distributive property of matrix multiplication: Using the properties and : Finally, we can solve for by dividing by 6:
Step 4: Final Answer:
The expression for matches option (D).

Concept:
The identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. The null matrix (or zero matrix) has all its elements equal to zero. Key properties:

• Adding a null matrix to any matrix does not change the matrix.
• The rank of the identity matrix is because all rows (and columns) are linearly independent.

Step 1: Add the matrices.
Let the identity matrix be The null matrix is Adding them:
Step 2: Find the rank of the resulting matrix.
Since the resulting matrix is still the identity matrix , all its rows (and columns) are linearly independent.

Concept:
If is an eigenvalue of a matrix , then the eigenvalue of is . This follows from the eigenvalue definition: Multiplying both sides by : Thus, the eigenvalues of are the squares of the eigenvalues of .
Step 1: Identify the eigenvalues of .
Given:
Step 2: Find eigenvalues of .
Square each eigenvalue:
Step 3: Write the eigenvalues of .