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Chapter 6: Problem 74

If \(A=\left[\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right],\) find \(\left(A^{-1}\right)^{-1}\)

Short Answer

Expert verified
\((A^{-1})^{-1} = A \) or in terms our given matrix, \(\left(\left[\begin{array}{ll}3 & 5 \ 2 & 4\end{array}\right]^{-1}\right)^{-1}=\left[\begin{array}{ll}3 & 5 \ 2 & 4\end{array}\right]\)

Step by step solution

01

Finding the Inverse of Matrix A

Start by finding the inverse of matrix \(A =\left(\begin{array}{ll}3 & 5 \ 2 & 4\end{array}\right)\). To find the inverse of a 2x2 matrix, interchanging the positions of \(a\) and \(d\), changing the signs of \(b\) and \(c\), and then dividing by the determinant (ad-bc) is the standard process. The formula for finding the inverse of the matrix is \(A^{-1}=\frac{1}{{ad - bc}}\times \left(\begin{array}{ll}d & -b \ -c & a\end{array}\right)\) where \(a=3\), \(b=5\), \(c=2\), and \(d=4\) for the given matrix \(A\). After calculation, we will get \(A^{-1}\).
02

Finding the Inverse of \(A^{-1}\)

Now we have the matrix \(A^{-1}\). Apply the same strategy to find the inverse of \(A^{-1}\), namely \( (A^{-1})^{-1} \). Using the method that we used in step 1, we find the inverse of \( A^{-1} \).
03

Conclusion

Compare the original matrix \(A\) with the double inverse \((A^{-1})^{-1}\). We will see that they are the same. It is a standard property of matrix operations that the double inverse of any matrix is the matrix itself. Hence, \((A^{-1})^{-1} = A\).

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Most popular questions from this chapter

Find two matrices \(A\) and \(B\) such that \(A B=B A\)

In Exercises \(5-8,\) find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{rr} x & 2 y \\ z & 9 \end{array}\right]=\left[\begin{array}{rr} 4 & 12 \\ 3 & 9 \end{array}\right] $$

In Exercises \(1-4\) a. Give the order of each matrix. b. If \(A=\left[a_{i j}\right],\) identify \(a_{32}\) and \(a_{23}\) or explain why identification is not possible. $$ \left[\begin{array}{rrr} -6 & 4 & -1 \\ -9 & 0 & \frac{1}{2} \end{array}\right] $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ A-X=4 B $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ 2 X+5 A=B $$
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