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Chapter 6: Chapter 6

Problem 166

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Find the inverse of A where $$ \mathrm{A}=\mid \begin{array}{rr} 2 & 3 \mid \\ \mid 3 & 5 \mid \end{array} $$

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The inverse of matrix A is: $$ \mathrm{A}^{-1}=\mid \begin{array}{rr} 5 & -3 \mid \\\ \mid -3 & 2 \mid \end{array} $$
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Find the determinant of matrix A

To find the determinant of a 2x2 matrix, we can use the following formula: \[ \det(A) = ad - bc \] For our given matrix A: $$ \mathrm{A}=\mid \begin{array}{rr} 2 & 3 \mid \\\ \mid 3 & 5 \mid \end{array} $$ a = 2, b = 3, c = 3, and d = 5. So, the determinant of A is: \[ \det(A) = (2)(5) - (3)(3) = 10 - 9 = 1 \]

Step 2: Find the inverse of matrix A using the formula

Since the determinant is non-zero, matrix A has an inverse. The formula for finding the inverse of a 2x2 matrix is as follows: \[ A^{-1} = \frac{1}{\det(A)}\mid \begin{array}{cc} d & -b\\ -c & a \end{array} \] Using the values of a, b, c, d, and the determinant calculated in step 1: \[ A^{-1} = \frac{1}{1}\mid \begin{array}{cc} 5 & -3\\ -3 & 2 \end{array} \]

Step 3: Simplify the inverse matrix

Since the determinant is equal to 1, the fraction in the formula doesn't affect the values of the matrix elements, and therefore the inverse of matrix A is: $$ \mathrm{A}^{-1}=\mid \begin{array}{rr} 5 & -3 \mid \\\ \mid -3 & 2 \mid \end{array} $$

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Most popular questions from this chapter

Chapter 6

Find the inverse of A where $$ \mathrm{A}=\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 5 & 4 \\ \mid 3 & 7 & 4 \end{array} $$
Chapter 6

Show that \(\mathrm{A}\) is not invertible where $$ \mathrm{A}=\begin{array}{llr} 1 & 6 & 4 \\ 2 & 4 & -1 \\ -1 & 2 & 5 \end{array} \mid $$
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 $$ \mathrm{A}=\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{array} $$
Chapter 6

Find the inverses of the following matrices. (1) $\quad \mathrm{A}=\mid \begin{array}{ll}3 & 1 \mid \\ \mid-1 & 6 \mid\end{array}$ (2) $\quad \mathrm{A}=\begin{array}{rrr} & 11 & -7 & -14 \\ & \mid 2 & 1 & -1 \\\ & \mid 1 & 3 & 4\end{array}$ (3) $\quad \begin{array}{rrr} & \mid 3 & 1 & 0 \\ & A=\mid 1 & -1 & 2 \mid \\\ & \mid 1 & 1 & 1 \mid\end{array}$
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