MTH1004 Linear Algebra Tutorial Class 4

Question 1

Question

Problem 1: (a) If $A$ is invertible, prove that a homogeneous system $Ax=0$ has unique solution the zero solution $x=0$.

Solution

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Question 2

Question

Problem 1: (b) If $A$ is invertible, what is the rank of $A$? Justify your answer.

Solution

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Question 3

Question

Problem 1: (c) Simplify the expression $(A^{-1}{B}^2{C}^{-2}{)}^{-1}$.

Solution

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Question 4

Question

Problem 1: (d) Prove that, if $A$ satisfies the matrix equation $4A^3-A^2-2I=O$, then $A$ is invertible.

Solution

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Question 5

Question

Problem 1: (e) Solve the matrix equation $A^{-1}X+B+3A=O$ in terms of the matrix $X$.

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Question 6

Question

Problem 2: Find the inverse of the matrix: (a) $A=\left(\begin{array}{cc}1& -2\\ 5& 7\end{array}\right)$ using the standard formula.

Solution

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Question 7

Question

Problem 2: Find the inverse of the matrix: (b) $R=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ using the standard formula.

Solution

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Question 8

Question

Problem 2: Find the inverse of the matrix: (c) $C=\left[\begin{array}{ccc}1& 1& 2\\ 1& 2& 1\\ 2& 1& 1\end{array}\right]$ using Gauss-Jordan elimination. By using its inverse $C^{-1}$, solve the system $Cx=u$, where $u=\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]$.

Solution

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