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Chapter 6: Chapter 6

Problem 156

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Find the inverse of $$ \begin{array}{ll} \mid 1 & 2 \\ \mid 3 & 7 \end{array} \mid $$

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The inverse of the given matrix is \(A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the determinant

Let the matrix be \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), in this case, \(a=1, b=2, c=3\), and \(d=7\). The determinant of A is calculated as: \[Det(A) = ad - bc\] \[Det(A) = (1)(7) - (2)(3) = 7 - 6 = 1\]

Step 2: Switch the elements at the main diagonal

Switch the a and d elements in the matrix A: \[A' = \begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix}\]

Step 3: Change the sign of the elements at the other diagonal

Change the sign of the b and c elements in the matrix A': \[A'' = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\]

Step 4: Divide the matrix by the determinant

Since the determinant is 1, we divide the matrix A'' by the determinant: \[A^{-1} = \frac{1}{1}A'' = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\] The inverse of the given matrix is: \[A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}\]

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Most popular questions from this chapter

Chapter 6

Let $\mathrm{A}=\begin{array}{ccc}1 & 2 & 3 \mid \\ \mid 1 & 3 & 2 \\ \mid 1 & 1 & 5\end{array}$ Show how we obtain the inverse of \(A\) by reducing the matrix [A: I] to a matrix of the form \([\mathrm{I}: \mathrm{B}]\).
Chapter 6

Find the inverse of $$ \mathrm{A}=\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{array} $$
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Let $\mathrm{A}=\begin{array}{ccc}\mid 1 & 1 & 1 \mid \\ \mid 0 & 1 & 0 \\\ \mid 0 & 1 & 1\end{array}$ Find the inverse of \(\mathrm{A}\).
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Find the inverse of A where $$ \mathrm{A}=\left|\begin{array}{rrr} 1 & 2 & -3 \\ 1 & -2 & 1 \\ \mid 5 & -2 & -3 \end{array}\right| $$
Chapter 6

Show that \(\mathrm{A}\) is not invertible where $$ \mathrm{A}=\begin{array}{llr} 1 & 6 & 4 \\ 2 & 4 & -1 \\ -1 & 2 & 5 \end{array} \mid $$
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