8. State the condition for the equation Ax = b to have a unique solution, where...

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8. State the condition for the equation Ax = b to have a unique solution, where A is a square matrix. Show that the system of linear equations x + ky + (k 1)z = 4 (k+1)x + (3k+1)x + (k-2)y + (k + 2)z = 2 (k-4)y + (2k + 5)z = 0 has a unique solution provided that k is not one of three certain real numbers, and find these three numbers. In each of the three exceptional cases determine whether the given equations have no solutions or infinitely many solutions. In each case of infinitely many solutions find the general solution. 8. State the condition for the equation Ax = b to have a unique solution, where A is a square matrix. Show that the system of linear equations x + ky + (k 1)z = 4 (k+1)x + (3k+1)x + (k-2)y + (k + 2)z = 2 (k-4)y + (2k + 5)z = 0 has a unique solution provided that k is not one of three certain real numbers, and find these three numbers. In each of the three exceptional cases determine whether the given equations have no solutions or infinitely many solutions. In each case of infinitely many solutions find the general solution.