Keywords : Row Echelon Form, System of Linear Equations,
Gauss elimination method, Linear Independent, and Linearly Dependent Problems with Solutions.
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(1) Reduce each of the following matrices into row echelon form and then find their ranks:
(a) \( \left[\begin{array}{llll} 1 & 2 & 1 & 0 \\ 2 & 4 & 8 & 6 \\ 0 & 0 & 5 & 8 \\ 3 & 6 & 6 & 3 \end{array}\right] \)
(b) \( \left[\begin{array}{ccc} 2 & 4 & 6 \\ -1 & 3 & 2 \\ 1 & 4 & 6 \\ 2 & 8 & 7 \end{array}\right] \)
(c) \( \left[\begin{array}{cccc} 1 & -2 & 5 & -3 \\ 2 & 3 & 1 & -4 \\ 3 & 8 & -3 & -5 \end{array}\right] \)
(d) \( \left[\begin{array}{ccccc} 0 & 0 & 2 & 2 & 0 \\ 1 & 3 & 2 & 4 & 1 \\ 2 & 6 & 2 & 6 & 2 \\ 3 & 9 & 1 & 10 & 6 \end{array}\right] \)
(2) Examine the following set of vectors over \(\mathbb{R}\) for linear dependence:
(a) \( \{(1,0,0),(0,1,0),(1,1,1),(-1,1,-1)\} \)
(b) \( \{(1,-1,1),(2,1,1),(8,1,5)\} \)
(c) \( \{(1,-1,2,4),(2,-1,5,7),(-1,3,1,-2)\} \)
(d) \( \{(1,2,1),(2,1,0),(1,-1,2)\} \)
(3) (a) Find the conditions on \(\alpha\) and \(\beta\) for which the matrix \( \left(\begin{array}{ccc} \alpha & 1 & 2 \\ 0 & 2 & \beta \\ 1 & 3 & 6 \end{array}\right) \) has:
(i) \(\text{rank} = 1\)
(ii) \(\text{rank} = 2\)
(iii) \(\text{rank} = 3\)
(b) For what values of \(\alpha\) and \(\beta\) is the following system consistent?
\( 2 x + 4 y + (\alpha + 3) z = 2, \)
\( x + 3 y + z = 2, \)
\( (\alpha – 2) x + 2 y + 3 z = \beta \)
(4) Solve the following system of linear equations by Gauss elimination method:
(a) \( x + 4 y – z = 4, \)
\( x + y – 6 z = -4, \)
\( 3 x – y – z = 1 \)
(b) \( x + y + z = -3, \)
\( 3 x + y – 2 z = -2, \)
\( 2 x + 4 y + 7 z = 7 \)
(c) \( x + 2 y + z = 2, \)
\( 3 x + y – 2 z = 1, \)
\( 2 x + 4 y + 2 z = 4 \)
(d) \( 2 \sin x – \cos y + 3 \tan z = 3, \)
\( 4 \sin x + 2 \cos y – 2 \tan z = 10, \)
\( 6 \sin x – 3 \cos y + \tan z = 9 \)
(5) Consider the following systems of linear equations:
(a) \( 2 x + 3 y + 5 z = 9, \)
\( 2 x + 3 y + r z = s, \)
\( 7 x + 3 y – 2 z = 8 \)
(b) \( x + y – z = 1, \)
\( 2 x + 3 y + \lambda z = 3, \)
\( x + \lambda y + 3 z = 2 \)
(c) \( \lambda x + y + z = p, \)
\( x + \lambda y + z = q, \)
\( x + y + \lambda z = r \)
Find the values of unknown constant(s) such that each of the above systems has:
(i) No solution
(ii) A unique solution
(iii) Infinitely many solutions
(6) Let \(P_{2}\) be the set of all polynomials of degree 2 or less. Use Gauss elimination method to find polynomial(s) \(f \in P_{2}\) such that \(f(0) = 1\), \(f(1) = 2\) and \(f(-1) = 6\).
(7) Find the values of \( k \) for which the following system of equations has:
(i) Trivial solution
(ii) Non-trivial solution
\( (3 k – 8) x + 3 y + 3 z = 0 \)
\( (k – 1) x + (3 k + 1) y + 2 k z = 0 \)
\( 3 x + (3 k – 8) y + 3 z = 0 \)
\( (k – 1) x + (4 k – 2) y + (k + 3) z = 0 \)
\( 3 x + 3 y + (3 k – 8) z = 0 \)
\( 2 x + (3 k + 1) y + 3 (k – 1) z = 0 \)
(8) By employing elementary row operations, find the inverse of the following matrices:
(a) \( \left(\begin{array}{ccc} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{array}\right) \)
(b) \( \left(\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 2 & 1 & 3 & 0 \\ 3 & 0 & 2 & 5 \\ 2 & 1 & 1 & 3 \end{array}\right) \)
(9) Suppose \( X, Y \in \mathbb{R}^{n} \), \( n > 1 \) are any two column matrices. Prove or disprove that the matrix \( A = X Y^{T} \) is invertible.
(10) Find the value of \( \theta \) for which the following system of equations has a non-trivial solution:
\( 2 (\sin \theta) x + y – 2 z = 0 \)
\( 3 x + 2 (\cos 2 \theta) y + 3 z = 0 \)
\( 5 x + 3 y – z = 0 \)
(11) (a) Let \( A \) be an \( n \times n \) matrix. If \( A \) is not invertible, then prove that there exists an \( n \times n \) matrix \( B \) such that \( A B = 0 \) but \( B \neq 0 \).
(b) Let \( A = \left[\begin{array}{cccc} 1 & 2 & -1 & 3 \\ -2 & 1 & 0 & 1 \\ 0 & 5 & -2 & 7 \\ -1 & 3 & -1 & 4 \end{array}\right] \) Find a \( 4 \times 4 \) matrix \( B \neq 0 \) such that \( A B = 0 \).
(12) Consider a \( 4 \times 5 \) matrix \( A = \left[\begin{array}{ccccc} 1 & 7 & -1 & -2 & -1 \\ 3 & 21 & 0 & 9 & 0 \\ 2 & 14 & 0 & 6 & 1 \\ 6 & 42 & -1 & 13 & 0 \end{array}\right] \)
(a) Find the row-reduced echelon form of \( A \).
(b) Find an invertible matrix \( P \) such that \( P A = \left[\begin{array}{ccccc} 1 & 7 & -1 & -2 & -1 \\ 0 & 0 & 3 & 15 & 3 \\ 0 & 0 & 2 & 10 & 3 \\ 0 & 0 & 5 & 25 & 6 \end{array}\right] \)
(c) Find the locus of the point \( (x, y, z) \in \mathbb{R}^{3} \) such that for each column vector \( Y = (x, y, z, 5)^{T} \) the equation \( A X = Y \) has a solution.
(d) If \( X = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5})^{T} \), then find the conditions on \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) such that \( A X = 0 \).
Answers
(1) (a) 3 (b) 3 (c) 2 (d) 3
(2) (a) LD (b) LD (c) LI (d) LI
\( \text{(3) (a) (i) Not possible} \quad \text{(ii) } \alpha = \frac{1}{3} \text{ or } \beta = 4 \quad \text{(iii) } \alpha \neq \frac{1}{3}, \beta \neq 4 \)
\( \text{(b) } \alpha = 3 \text{ and } \beta = 1 \text{ ; } \quad \text{or } \quad \alpha = -2 \text{ and } \beta = 6 \text{ ; } \quad \text{or } \quad \alpha \neq 3, -2 \)
(4) (a) \((1,1,1)\) (b) No solution (c) Infinite solutions (d) No solution
(5) (a) (i) \( r = 5, s \neq 9 \quad \text{(ii) } r \neq 5, s \in \mathbb{R} \quad \text{(iii) } r = 5, s = 9 \)
\( \text{(b) (i) } \lambda = -3 \quad \text{(ii) } \lambda \neq -3, 2 \quad \text{(iii) } \lambda = 2 \)
\( \text{(c) (i) } \lambda = 1 \text{ and } p + q – 2r \neq 0 \text{ OR } \lambda = 1 \text{ and } q \neq r \text{ OR } \lambda = 1 \text{ and } r \neq p \text{ OR } \lambda = 1 \text{ and } p \neq q \text{ OR } \lambda = -2 \text{ and } p + q + r \neq 0 \text{ and } q \neq r \)
\( \text{(ii) } \lambda \neq 1, -2 \)
\( \text{(iii) } \lambda = 1 \text{ and } p = q = r \text{ OR } \lambda = -2 \text{ and } p + q + r = 0 \)
(6) \( f(x) = 3 x^{2} – 2 x + 1 \)
(7) (a) (i) \( k \neq \frac{2}{3}, \frac{11}{3} \quad \text{(ii) } k = \frac{2}{3} \text{ or } \frac{11}{3} \)
\( \text{(b) (i) } k \neq 0, 3 \quad \text{(ii) } k = 0 \text{ or } 3 \)
(8) (a) \( \left[\begin{array}{lll} 3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 2 & 5 \end{array}\right] \)
(b) \( \frac{1}{4} \left[\begin{array}{cccc} -16 & 4 & -4 & 12 \\ 5 & -1 & -1 & 0 \\ 9 & -1 & 3 & -8 \\ 6 & -2 & 2 & -4 \end{array}\right] \)
(10) \( n \pi + (-1)^{n} \frac{\pi}{6} \text{ or } n \pi + (-1)^{n} \sin^{-1}(9 – \sqrt{161}) / 4, \, n = 0, 1, 2, \ldots \)
(11) (b) \( \left[\begin{array}{cccc} 1 & -1 & 1 & 0 \\ 2 & -7 & 2 & -5 \\ 5 & 0 & 5 & 5 \\ 0 & 5 & 0 & 5 \end{array}\right] \) (This is just one solution. The matrix \( B \) is not unique).
(12) (a) \( \left[\begin{array}{lllll} 1 & 7 & 0 & 3 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] \)
(b) \( \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ -3 & 1 & 0 & 0 \\ -2 & 0 & 1 & 0 \\ -6 & 0 & 0 & 1 \end{array}\right] \)
(c) \( x + y + z = 5 \)
(d) \( x_{1} + 7 x_{2} + 3 x_{4} = 0, \quad x_{3} + 5 x_{4} = 0, \quad x_{5} = 0 \)