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(Solved): 1. Solve the following system using :a) Gauss-Jordan algorithm, b) the invers ...
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1. Solve the following system using :a) Gauss-Jordan algorithm, b) the inverse of the corresponding matrix, c) Cramer's Rule \[ \begin{array}{r} 2 x_{1}-3 x_{2}+4 x_{3}=3 \\ x_{1}-3 x_{2}+x_{3}=-1 \\ 3 x_{1}+2 x_{2}+5 x_{3}=10 \end{array} \] 2. Find the subspaces of vector space V ColA, NulA and RowA, bases of these subspaces. Identify dimensions of these subspaces, rankA and relationship between them \[ \left[\begin{array}{ccccc} 1 & 2 & -4 & 3 & 3 \\ 5 & 10 & -9 & -7 & 8 \\ 4 & 8 & -9 & -2 & 7 \\ -2 & -4 & 5 & 0 & -6 \end{array}\right] \] 4. Find \( A^{3} \), if the matrix \( \mathrm{A} \) is \[ \left[\begin{array}{ccc} 1 & 2 & -4 \\ 2 & -5 & 2 \\ -4 & 2 & 1 \end{array}\right] \] 1. (5 points) Solve the system by means of Gauss-Jordan algorithm: \[ \left\{\begin{array}{c} x_{1}+4 x_{2}-2 x_{3}+8 x_{4}=12 \\ x_{2}-7 x_{3}+2 x_{4}=-4 \\ 5 x_{3}-x_{4}=7 \end{array}\right. \] 2. (15 points) Solve the following system using : a) the inverse of the corresponding matrix, b) Cramer's rule \[ \begin{array}{l} 2 x_{1}-x_{2}+x_{3}=1 \\ 6 x_{1}-2 x_{3}=4 \\ 8 x_{1}-x_{2}+5 x_{3}=11 \end{array} \] 3. (10 points) Find Col \( A, N u l ~ A \), Row \( A \), basis of \( \operatorname{Col~A} \), basis of Nul \( A \) and basis of Row \( A \) subspaces of the given matrix. Define dimensions of these subspaces, find rank \( A \) and relationship between them. \[ A=\left[\begin{array}{ccccc} 1 & 2 & -5 & 11 & -3 \\ 2 & 4 & -5 & 15 & 2 \\ 1 & 2 & 0 & 4 & 5 \\ 3 & 6 & -5 & 19 & -2 \end{array}\right] \] 4. (10 points) Find \( A^{3} \), if \[ A=\left[\begin{array}{cc} -2 & 12 \\ -1 & 5 \end{array}\right] \]
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