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Chapter 4: Problem 336

If \(\mathrm{A}=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|\) \(8 \mathrm{~A}^{-4}=\) (a) \(145 \overline{\mathrm{A}^{-1}-27 \mathrm{I}}\) (b) \(27 \mathrm{I}-145 \mathrm{~A}^{-1}\) (c) \(29 \mathrm{~A}^{-1}+9 \mathrm{I}\) (d) \(145 \mathrm{~A}^{-1}+27 \mathrm{I}\)

Short Answer

Expert verified
The short answer is: (b) \(27 \text{I} - 145 A^{-1}\).

Step by step solution

01

Find A^(-1)

To find the inverse of a 2x2 matrix, we can use the formula: \(A^{-1} = \dfrac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\), where \(\text{det}(A) = ad - bc\). We have A: \[A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\] Let's find the determinant and the inverse: \(\text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2\) Now, let's find \(A^{-1}\) : \[A^{-1} = \dfrac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix}\]
02

Find A^(-4)

To find \(A^{-4}\), we multiply \(A^{-1}\) by itself four times. \(A^{-4} = A^{-1}A^{-1}A^{-1}A^{-1}\) Note that \((A^{-1})^2 = A^{-1} A^{-1}\). To find \((A^{-1})^2\), let's multiply \(A^{-1}\) by itself: \[\begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix} \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix} = \begin{bmatrix} 1/2 & -1/2 \\ -1/2 & 1/2 \end{bmatrix}\] Now let's find \(A^{-4}\): \[\begin{bmatrix} 1/2 & -1/2 \\ -1/2 & 1/2 \end{bmatrix} \begin{bmatrix} 1/2 & -1/2 \\ -1/2 & 1/2 \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 2 & -5 \end{bmatrix}\]
03

Multiply A^(-4) by 8

Next, let's multiply the result \((A^{-4})\) by 8: \(8A^{-4} = 8 \begin{bmatrix} 1 & -2 \\ 2 & -5 \end{bmatrix} = \begin{bmatrix} 8 & -16 \\ 16 & -40 \end{bmatrix}\)
04

Compare to given options

Now, we have to compare the given options to what we found as \(8A^{-4}\): (a) \(145 \overline{A^{-1}-27I}\) does not match. (b) \(27I - 145A^{-1} = \begin{bmatrix} 27 & 0 \\ 0 & 27 \end{bmatrix} - 145 \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix} = \begin{bmatrix} 8 & -16 \\ 16 & -40 \end{bmatrix}\), which matches our result. (c) and (d) do not match our result. Therefore, the correct answer is (b) \(27 \text{I} - 145 A^{-1}\).

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