Linear Algebra

21. The 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

0
either 0 or 1
one of 0, 1 or -1
any real number other than 5

Detailed SolutionThe following system of equations x1 + x2 + 2×3 = 1 x1 + 2×3 + 3×3 = 2 x1 + 4×2 + ax3 = 4 has

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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

22. The 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

5
7
9
18

Detailed SolutionThe sum

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

23. The 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

-3, -4
-1, -2
-1, -3
-2, -4

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href="https://exam.pscnotes.com/mcq/the-eigen-values-of-a-2-a%c2%97-2-matrix-x-are-2-and-3-the-eigen-values-of-the-matrix-x-i-x-5i-are-a-3-4-b-1-2-c-1-3-d-2-4/#more-20044">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

24. If 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

p - q + r = 0 or p = q = -r
p + q - r = 0 or p = -q = r
p + q + r = 0 or p = q = r
p - q + r = 0 or p = -q = -r

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p = q = -r B. p + q – r = 0 or p = -q = r
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C. p + q + r = 0 or p = q = r D. p – q + r = 0 or p = -q = -r" class="read-more button" href="https://exam.pscnotes.com/mcq/if-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/#more-20045">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

25. Let 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

0, 0, 0
0, 0, 1
0, 1, 1
1, 1, 1

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0 576 512"> Subscribe on YouTube
values of N are A. 0, 0, 0 B. 0, 0, 1 C. 0, 1, 1 D. 1, 1, 1" class="read-more button" href="https://exam.pscnotes.com/mcq/let-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/#more-20043">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

26. Consider 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

x = 4, y = 10
x = 5, y = 8
x = -3, y = 9
x = -4, y = 10

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\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" class="read-more button" href="https://exam.pscnotes.com/mcq/consider-the-following-matrix-texta-left-beginarray20c-23-textxtexty-endarray-right-if-the-eigen-values-of-a-are-4-and-8-then-a-x-4-y-10/#more-20042">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

27. The 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. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&{ – 2} \\ { – 5}&{ – 1} \end{array}} \right]\]

”[ rac{1}{3}left[
\]” 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″]

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C. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} 7&{ – 2} \\ { – 5}&1 \end{array}} \right]\] D. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&{ – 2} \\ { – 5}&{ – 1} \end{array}} \right]\]" class="read-more button" href="https://exam.pscnotes.com/mcq/the-inverse-of-the-2-a%c2%97-2-matrix-left-beginarray20c-12-57-endarray-right-is-a-frac13left-beginarray20c-72-5-1-endar/#more-20040">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. \[\frac{1}{3}\left[ {\begin{array}{*{20}{c}} { – 7}&{ – 2} \\ { – 5}&{ – 1} \end{array}} \right]\]

28. How 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

infinitely many
two distinct solutions
unique
none

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64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z"/> Subscribe on YouTube 3y = 3 A. infinitely many B. two distinct solutions C. unique D. none" class="read-more button" href="https://exam.pscnotes.com/mcq/how-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/#more-20041">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

29. Which 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. \[\left| {\begin{array}{*{20}{c}} 2&{{\text{x}} + {\text{y}}}&{{{\text{x}}^2} + {{\text{y}}^2}} \\ 2&{{\text{y}} + {\text{z}}}&{{{\text{y}}^2} + {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]

”[left|
” 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″]

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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. \[\left| {\begin{array}{*{20}{c}} 2&{{\text{x}} + {\text{y}}}&{{{\text{x}}^2} + {{\text{y}}^2}} \\ 2&{{\text{y}} + {\text{z}}}&{{{\text{y}}^2} + {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]" class="read-more button" href="https://exam.pscnotes.com/mcq/which-one-of-the-following-does-not-equal-left-beginarray20c-1textxtextx2-1textytexty2-1textztextz2-endarray/#more-20039">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. \[\left| {\begin{array}{*{20}{c}} 2&{{\text{x}} + {\text{y}}}&{{{\text{x}}^2} + {{\text{y}}^2}} \\ 2&{{\text{y}} + {\text{z}}}&{{{\text{y}}^2} + {{\text{z}}^2}} \\ 1&{\text{z}}&{{{\text{z}}^2}} \end{array}} \right|\]

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

always zero
always pure imaginary
either zero or pure imaginary
always real

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always real" class="read-more button"
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href="https://exam.pscnotes.com/mcq/the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real/#more-20038">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

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