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2. For the given matrices A find A-¹ if it exists and verify that AA-¹ = A-¹A=I. If A-¹ does not exist explain why. (a) A = ·(₁ (b) A = (c) A = ( 2 -1 0 1 02 ¹) -1 = (1+i)

2. For the given matrices A find A-¹ if it exists and verify that AA-¹ = A-¹A=I. If A-¹ does not exist explain why. (a) A = ·(₁ (b) A = (c) A = ( 2 -1 0 1 02 ¹) -1 = (1+i)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem Statement **2.** For the given matrices \( A \), find \( A^{-1} \) if it exists and verify that \( AA^{-1} = A^{-1}A = I \). If \( A^{-1} \) does not exist, explain why. #### Matrices: (a) \( A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} \) (b) \( A = \begin{pmatrix} 0 & 1 \\ 0 & 2 \end{pmatrix} \) (c) \( A = \begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} \) ### Explanation - **Matrix (a)** is a \(2 \times 2\) matrix that needs to be tested for invertibility. The condition for a matrix to be invertible is that its determinant should not be zero. - **Matrix (b)** is also a \(2 \times 2\) matrix. Here, the first row begins with a zero, which might indicate that it is singular. The determinant helps confirm if it's invertible. - **Matrix (c)** is a \(2 \times 2\) matrix involving a variable \( c \). The general form suggests it may have special properties, such as a quick way to find an inverse, depending on the value of \( c \).
Transcribed Image Text:### Problem Statement **2.** For the given matrices \( A \), find \( A^{-1} \) if it exists and verify that \( AA^{-1} = A^{-1}A = I \). If \( A^{-1} \) does not exist, explain why. #### Matrices: (a) \( A = \begin{pmatrix} 2 & -1 \\ -1 & 2 \end{pmatrix} \) (b) \( A = \begin{pmatrix} 0 & 1 \\ 0 & 2 \end{pmatrix} \) (c) \( A = \begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} \) ### Explanation - **Matrix (a)** is a \(2 \times 2\) matrix that needs to be tested for invertibility. The condition for a matrix to be invertible is that its determinant should not be zero. - **Matrix (b)** is also a \(2 \times 2\) matrix. Here, the first row begins with a zero, which might indicate that it is singular. The determinant helps confirm if it's invertible. - **Matrix (c)** is a \(2 \times 2\) matrix involving a variable \( c \). The general form suggests it may have special properties, such as a quick way to find an inverse, depending on the value of \( c \).
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