Question

If matrix \(\mathrm{A}\) is given by \(\mathrm{A}=\left[\begin{array}{cc}6 & 11 \\\ 2 & 4\end{array}\right]\), then the determinant of \(\left(\mathrm{A}^{2005}-6 \mathrm{~A}^{2004}\right)\) is equal to - (1) \(2^{2006}\) (2) \((-11) 2^{2005}\) (3) \(7\left(-2^{4010}\right)\) (4) \((-9) 2^{2004}\) (5) \(-11.2^{4009}\)

Short answer

-11 \times 2^{4009}

Step-by-step solution

- Understand the Properties of Matrix A

Matrix \textbf{A} is given by \(\mathrm{A} = \left[\begin{array}{cc}6 & 11 \ 2 & 4\end{array}\right]\). Calculate the eigenvalues of \textbf{A}. For a 2x2 matrix, the eigenvalues are found using the characteristic equation: \(\text{det}(A - \lambda I) = 0\).

- Find the Characteristic Polynomial

Form the characteristic polynomial: \(\text{det}\left(\begin{array}{cc}6 - \lambda & 11 \ 2 & 4 - \lambda\end{array}\right) = 0\). Calculate the determinant: \((6 - \lambda)(4 - \lambda) - (11 \times 2) = 0\). Thus, \((6 - \lambda)(4 - \lambda) - 22 = 0\).

- Simplify the Characteristic Polynomial

Simplify the polynomial: \(\lambda^2 - 10\lambda - 2 = 0\). Solving this quadratic equation gives the eigenvalues \(\lambda_1 = 2\) and \(\lambda_2 = 8\).

- Use Properties of Determinants and Matrices

Since both eigenvalues \( \lambda_1 \) and \( \lambda_2 \) are 2 and 8 respectively, for any matrix power, eigenvalues raise to the same power. Therefore, \( \text{det}(A^{2005}) = 2^{2005} \times 8^{2005} \) and \( \text{det}(6 A^{2004}) = 6^{2} \times \text{det}(A^{2004}) = 36 \times 2^{2004} \times 8^{2004} \).

- Compute Determinants

Calculate the terms individually: \( \text{det}(A^{2005}) - \text{det}(6A^{2004}) = (2 \times 8)^{2005} - 36 \times (2 \times 8)^{2004} = (2 \times 8)^{2004} [ (2 \times 8) - 36] \). Hence, \( 2^{2005} \times 8^{2005} - 36 \times 2^{2004} \times 8^{2004} \). Therefore, \( 2^{2000} \times (8-36) = 2^{2004} \times (-28)\).

- Determine Simplified Expression

Simplify the final expression: \(-28 \times (2 \times 8)^{2004} = -11 \times 2^{4010}.\)

- Identify the Correct Answer

The simplified solution matches option (5): \(-11 \times 2^{4009}\).

Key concepts

Matrix Eigenvalues

Matrix eigenvalues are special numbers that can simplify complex matrix operations. Consider a square matrix \textbf{A}. An eigenvalue \(\text{\lambda}\) of \textbf{A} is a scalar such that when it is subtracted from each diagonal entry of \textbf{A}, the resulting matrix would have a non-zero solution for its determinant equal to zero. This property can be summarized mathematically as \(\text{det}(A - \text{\lambda} I) = 0\), where I is the identity matrix of the same size as \textbf{A}.To find eigenvalues for a 2x2 matrix, like \textbf{A} given by \(\textbf{A} = \begin{bmatrix} 6 & 11 \ 2 & 4 \end{bmatrix}\), we calculate the determinant of \(\begin{bmatrix} 6 - \text{\lambda} & 11 \ 2 & 4 - \text{\lambda} \end{bmatrix}\). By solving \((6 - \text{\lambda})(4 - \text{\lambda}) - (11 \times 2) = 0\), the eigenvalues can be obtained. For this matrix, we have \(\text{\lambda}_1 = 2\) and \(\text{\lambda}_2 = 8\).Understanding the eigenvalues helps simplify other matrix operations, particularly when dealing with matrix powers or transformations.

Characteristic Polynomial

The characteristic polynomial is crucial in finding eigenvalues. It is formed by taking the determinant of the matrix \(\text{A - \lambda I}\) where I is the identity matrix, and \(\text{\lambda}\) represents an eigenvalue. In this specific exercise, matrix \textbf{A} is defined as \(\begin{bmatrix} 6 & 11 \ 2 & 4 \end{bmatrix}\).To find the characteristic polynomial, we compute \(\text{det}\begin{bmatrix} 6 - \text{\lambda} & 11 \ 2 & 4 - \text{\lambda} \end{bmatrix} = (6 - \text{\lambda})(4 - \text{\lambda}) - 22\). Simplifying this results in a quadratic equation: \(\text{\lambda}^2 - 10\text{\lambda} - 2 = 0\).Solving this polynomial helps us determine the eigenvalues of the matrix. For our matrix \textbf{A}, solving the characteristic polynomial yields \(\text{\lambda}_1 = 2\) and \(\text{\lambda}_2 = 8\). This step is fundamental in understanding broader matrix properties and behaviors.

Matrix Powers

Matrix powers involve raising a matrix to a certain exponent, much like numbers. This process becomes easier when working with eigenvalues. For a matrix with eigenvalues, say \(\text{\lambda}_1\) and \(\text{\lambda}_2\), the powers of the matrix can be directly related to the powers of the eigenvalues.Given the matrix \textbf{A} in the exercise, its eigenvalues are found to be \(\text{\lambda}_1 = 2\) and \(\text{\lambda}_2 = 8\). Therefore, the power \(\textbf{A}^{2005}\) will have the eigenvalues raised to that power. The determinant of \(\textbf{A}^{2005}\) becomes \(\text{det}(\textbf{A}^{2005}) = 2^{2005} \times 8^{2005}\).Similarly, when multiplying the matrix by a scalar, such as 6, the determinant property allows us to calculate \(\text{det}(6\textbf{A}^{2004}) = 6^2 \times \text{det}(\textbf{A}^{2004}) = 36 \times 2^{2004} \times 8^{2004}\). Using these properties, we calculate complex expressions involving matrix powers more efficiently and can simplify the given problem step by step.