Linear Algebra

Detailed SolutionThe following system of equations x1 + x2 + 2×3 = 1 x1 + 2×3 + 3×3 = 2 x1 + 4×2 + ax3 = 4 has a unique solution. The only possible value(s) for a is/are A. 0 B. either 0 or 1 C. one of 0, 1 or -1 D. any real number other than 5

Detailed SolutionThe sum of the eigen values of the matrix given below is \[\left[ {\begin{array}{*{20}{c}} 1&2&3 \\ 1&5&1 \\ 3&1&1 \end{array}} \right].\] A. 5 B. 7 C. 9 D. 18

Detailed SolutionThe eigen values of a (2 × 2) matrix X are -2 and -3. The eigen values of the matrix (X + $$I$$) (X + 5$$I$$) are A. -3, -4 B. -1, -2 C. -1, -3 D. -2, -4

Detailed SolutionIf the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? A. p – q + r = 0 or p = q = -r B. p + q – r = 0 or p = -q = r C. p + q + r = 0 or p = q = r D. p – q + r = 0 or p = -q = -r

Detailed SolutionLet N be a 3 by 3 matrix with real number entries. The matrix N is such that N2 = 0. The eigen values of N are A. 0, 0, 0 B. 0, 0, 1 C. 0, 1, 1 D. 1, 1, 1

Detailed SolutionConsider the following matrix. \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&3 \\ {\text{x}}&{\text{y}} \end{array}} \right]\] If the eigen values of A are 4 and 8, then A. x = 4, y = 10 B. x = 5, y = 8 C. x = -3, y = 9 D. x = -4, y = 10

\]” option2=”\[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} 7&2 \\ 5&1 \end{array}} \right]\]” option3=”\[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} 7&{ – 2} \\ { – 5}&1 \end{array}} \right]\]” option4=”\[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&{ – 2} \\ { – 5}&{ – 1} \end{array}} \right]\]” correct=”option3″]

Detailed SolutionThe inverse of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ 5&7 \end{array}} \right]\] is A. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&2 \\ 5&{ – 1} \end{array}} \right]\] B. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} 7&2 \\ 5&1 \end{array}} \right]\] C. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} 7&{ – 2} \\ { – 5}&1 \end{array}} \right]\] D. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&{ – 2} \\ { – 5}&{ – 1} \end{array}} \right]\]

Detailed SolutionHow many solutions does the following system of linear equations have? -x + 5y = -1; x – y = 2; x + 3y = 3 A. infinitely many B. two distinct solutions C. unique D. none

” option2=”\[\left| {\begin{array}{*{20}{c}} 1&{{\text{x}} + 1}&{{{\text{x}}^2} + 1} \\ 1&{{\text{y}} + 1}&{{{\text{y}}^2} + 1} \\ 1&{{\text{z}} + 1}&{{{\text{z}}^2} + 1} \end{array}} \right|\]” option3=”\[\left| {\begin{array}{*{20}{c}} 0&{{\text{x}} – {\text{y}}}&{{{\text{x}}^2} – {{\text{y}}^2}} \\ 0&{{\text{y}} – {\text{z}}}&{{{\text{y}}^2} – {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]” option4=”\[\left| {\begin{array}{*{20}{c}} 2&{{\text{x}} + {\text{y}}}&{{{\text{x}}^2} + {{\text{y}}^2}} \\ 2&{{\text{y}} + {\text{z}}}&{{{\text{y}}^2} + {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]” correct=”option3″]

Detailed SolutionWhich one of the following does NOT equal \[\left| {\begin{array}{*{20}{c}} 1&{\text{x}}&{{{\text{x}}^2}} \\ 1&{\text{y}}&{{{\text{y}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|?\] A. \[\left| {\begin{array}{*{20}{c}} 1&{{\text{x}}\left( {{\text{x}} + 1} \right)}&{{\text{x}} + 1} \\ 1&{{\text{y}}\left( {{\text{y}} + 1} \right)}&{{\text{y}} + 1} \\ 1&{{\text{z}}\left( {{\text{z}} + 1} \right)}&{{\text{z}} + 1} \end{array}} \right|\] B. \[\left| {\begin{array}{*{20}{c}} 1&{{\text{x}} + 1}&{{{\text{x}}^2} + 1} \\ 1&{{\text{y}} + 1}&{{{\text{y}}^2} + 1} \\ 1&{{\text{z}} + 1}&{{{\text{z}}^2} + 1} \end{array}} \right|\] C. \[\left| {\begin{array}{*{20}{c}} 0&{{\text{x}} – {\text{y}}}&{{{\text{x}}^2} – {{\text{y}}^2}} \\ 0&{{\text{y}} – {\text{z}}}&{{{\text{y}}^2} – {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\] D. \[\left| {\begin{array}{*{20}{c}} 2&{{\text{x}} + {\text{y}}}&{{{\text{x}}^2} + {{\text{y}}^2}} \\ 2&{{\text{y}} + {\text{z}}}&{{{\text{y}}^2} + {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]

Detailed SolutionThe eigen values of a skew-symmetric matrix are A. always zero B. always pure imaginary C. either zero or pure imaginary D. always real