Linear Algebra — Part 1 MCQs with Explanations

1 is the correct answer to “If $${\left[ {1,\,0,\, - 1} \right]^{\text{T}}}$$ is an eigen vector of the following matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0 \\ { - 1}&2&{ - 1} \\ 0&{ - 1}&1 \end{array}} \right]\] then corresponding eigen value is”. This question is standard in competitive examination syllabi for Linear Algebra.

3, -1 + 3j, -1 - 3j is the correct answer to “The eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} { - 1}&3&5 \\ { - 3}&{ - 1}&6 \\ 0&0&3 \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

512√2 and -512√2 is the correct answer to “Let A be the 2 × 2 matrix with elements a 11 = a 12 = a 21 = +1 and a 22 = -1. Then the eigen values of the matrix A 19 are”. This question is standard in competitive examination syllabi for Linear Algebra.

one is the correct answer to “How many of the following matrices have an eigen value 1? \[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right],\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 1&1 \end{array}} \right]{\text{ and }}\left[ {\begin{array}{*{20}{c}} { - 1}&0 \\ 1&{ - 1} \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

AT = -A is the correct answer to “A real square matrix A is called skew-symmetric if”. This question is standard in competitive examination syllabi for Linear Algebra.

[5 -3 1] is the correct answer to “If \[{\text{R}} = \left[ {\begin{array}{*{20}{c}} 1&0&{ - 1} \\ 2&1&{ - 1} \\ 2&3&2 \end{array}} \right],\] then top row of R -1 is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{7}{{20}}\] is the correct answer to “Let, \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&{ - 0.1} \\ 0&3 \end{array}} \right]\] and \[{{\text{A}}^{ - 1}} = \left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&{\text{a}} \\ 0&{\text{b}} \end{array}} \right].\] Then (a + b) =”. This question is standard in competitive examination syllabi for Linear Algebra.

\[{\text{P}} \equiv {\text{Q}} \equiv {\text{R}} \equiv {\text{S}}\] is the correct answer to “We have a set of 3 linear equations in 3 unknowns. 'X \[ \equiv \] Y' means X and Y are equivalent statements and 'X \[\not \equiv \] Y' means X and Y are not equivalent statements. P : There is a unique solution. Q : The equations are linearly independent. R : All eigen values of the coefficient matrix are nonzero. S : The determinant of the coefficient matrix is nonzero. Which one of the following is TRUE”. This question is standard in competitive examination syllabi for Linear Algebra.

1, 5, ±j6 is the correct answer to “The elgen values of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&5 \\ 0&5&6 \\ 0&{ - 6}&5 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Linear Algebra.

\[{\lambda ^2} - 4\lambda - 5 = 0\] is the correct answer to “Consider the following simultaneous equations (with c 1 and c 2 being constants): 3x 1 + 2x 2 = c 1 4x 1 + x 2 = c 2 The characteristics equation for these simultaneous equations is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 2 \\ 5 \\ 0 \end{array}} \right]\] is the correct answer to “For the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 3&{ - 2}&2 \\ 0&{ - 2}&1 \\ 0&0&1 \end{array}} \right],\] one of the eigen values is equal to -2. Which of the following is an eigen vector”. This question is standard in competitive examination syllabi for Linear Algebra.

1 and 6 is the correct answer to “What are the eigen values of the following 2 × 2 matrix? \[\left[ {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 4}&5 \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

1, 4, 4 is the correct answer to “Eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 3&{ - 1}&{ - 1} \\ { - 1}&3&{ - 1} \\ { - 1}&{ - 1}&3 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} {38}&{28} \\ {32}&{56} \end{array}} \right]\] is the correct answer to “If \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&5 \\ 6&2 \end{array}} \right]\] and \[{\text{B}} = \left[ {\begin{array}{*{20}{c}} 3&7 \\ 8&4 \end{array}} \right],\,{\text{A}}{{\text{B}}^{\text{T}}}\] is equal to”. This question is standard in competitive examination syllabi for Linear Algebra.

1 is the correct answer to “The number of linearly independent eigen vectors of \[\left[ {\begin{array}{*{20}{c}} 2&1 \\ 0&2 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

0, 1, -1 is the correct answer to “A 3 × 3 matrix P is such that, P 3 = P. Then the eigen values of P are”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left\{ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right\}\] is the correct answer to “One of the eigen vectors of matrix is \[\left[ {\begin{array}{*{20}{c}} { - 5}&2 \\ { - 9}&6 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

6 is the correct answer to “For the matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 2&4 \end{array}} \right]\] the eigen value corresponding to the eigen vector \[\left[ {\begin{array}{*{20}{c}} {101} \\ {101} \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

3.48 and 13.53 is the correct answer to “The eigen values of matrix \[\left[ {\begin{array}{*{20}{c}} 9&5 \\ 5&8 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Linear Algebra.

15 is the correct answer to “Consider the 5 × 5 matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&2&3&4&5 \\ 5&1&2&3&4 \\ 4&5&1&2&3 \\ 3&4&5&1&2 \\ 2&3&4&5&1 \end{array}} \right]\] It is given that A has only one real eigen value. Then the real eigen value of A is”. This question is standard in competitive examination syllabi for Linear Algebra.

2 is the correct answer to “The rank of the matrix \[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \\ 1&0&2 \\ 3&6&6 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left\{ {\begin{array}{*{20}{c}} 2 \\ { - 1} \end{array}} \right\}\] is the correct answer to “One of the eigen vectors of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right]\] is”. This question is standard in competitive examination syllabi for Linear Algebra.

2 is the correct answer to “The lowest Eigen value of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&3 \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative is the correct answer to “Which one of the following statements is TRUE about every n × n matrix with only real eigen values”. This question is standard in competitive examination syllabi for Linear Algebra.

multiple solutions is the correct answer to “The equation \[\left[ {\begin{array}{*{20}{c}} 2&{ - 2} \\ 1&{ - 1} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{{\text{x}}_1}} \\ {{{\text{x}}_2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]\] has”. This question is standard in competitive examination syllabi for Linear Algebra.

no solution is the correct answer to “Consider the system of simultaneous equations x + 2y + z = 6 2x + y + 2z = 6 x + y + z = 5 This system has”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\lambda = 6,\,\mu \ne 20\] is the correct answer to “The system of equations x + y + z = 6 x + 4y + 6z = 20 x + 4y + \[\lambda \]z = \[\mu \] has NO solution for values of \[\lambda \] and \[\mu \] given by”. This question is standard in competitive examination syllabi for Linear Algebra.

2, 14; x1, x2 is the correct answer to “Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x 1 and x 2 respectively. Then the eigen values and eigen vectors of the matrix A 2 - 3A + 4$$I$$ would, respectively, be”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&{ - 1} \end{array}} \right]\] is the correct answer to “The figure shows a shape ABC and its mirror image A 1 B 1 C 1 across the horizontal axis (X-axis). The coordinate transformation matrix that maps ABC to A 1 B 1 C 1 is”. This question is standard in competitive examination syllabi for Linear Algebra.

1 is the correct answer to “If the entries in each column of a square matrix M add up to 1, then an eiqen value of M is”. This question is standard in competitive examination syllabi for Linear Algebra.

915 is the correct answer to “The sum of Eigen values of matrix, [M] is where \[\left[ {\text{M}} \right] = \left[ {\begin{array}{*{20}{c}} {215}&{650}&{795} \\ {655}&{150}&{835} \\ {485}&{355}&{550} \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

4 is the correct answer to “The set of equations x + y + z = 1 ax - ay + 3z = 5 5x - 3y + az = 6 has infinite solution if a =”. This question is standard in competitive examination syllabi for Linear Algebra.

For matrix Am, m being a positive integer, (λim, xim) will be the eigen-pair for all i is the correct answer to “Consider the system of equations A (n × n) X (n × 1) = λ (n × 1) where, λ is a scalar. Let (λ i , x i ) be an eigen-pair of an eigen value and its corresponding eigen vector for real matrix A. Let $$I$$ be a (n × n) unit matrix. Which one of the following statement is NOT correct”. This question is standard in competitive examination syllabi for Linear Algebra.

5 - j is the correct answer to “The value of x for which all the eigen-values of the matrix given below are real is \[\left[ {\begin{array}{*{20}{c}} {10}&{5 + {\text{j}}}&4 \\ {\text{x}}&{20}&2 \\ 4&2&{ - 10} \end{array}} \right]\]”. This question is standard in competitive examination syllabi for Linear Algebra.

infinitely many solutions is the correct answer to “The system of linear equations \[\left[ {\begin{array}{*{20}{c}} 2&1&3 \\ 3&0&1 \\ 1&2&5 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{a}} \\ {\text{b}} \\ {\text{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 5 \\ { - 4} \\ {14} \end{array}} \right]\] has”. This question is standard in competitive examination syllabi for Linear Algebra.

120° is the correct answer to “The matrix \[{\text{P}} = \left[ {\begin{array}{*{20}{c}} 0&0&1 \\ 1&0&0 \\ 0&1&0 \end{array}} \right]\] rotates a vector about the axis \[\left[ {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right]\] by angle of”. This question is standard in competitive examination syllabi for Linear Algebra.

At least 15 is the correct answer to “If a square matrix of order 100 has exactly 15 distinct eigen values, the degree of the minimal polynomial is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\left[ {\begin{array}{*{20}{c}} 0&1 \\ { - 2}&{ - 3} \end{array}} \right]\] is the correct answer to “A matrix has eigen values -1 and -2. The corresponding eigen vectors are \[\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ { - 2} \end{array}} \right]\] respectively. The matrix is”. This question is standard in competitive examination syllabi for Linear Algebra.

CDA is the correct answer to “Let A, B, C, D be n × n matrices, each with nonzero determinant, If ABCD = $$I$$, then B -1 is”. This question is standard in competitive examination syllabi for Linear Algebra.

\[\frac{1}{2}\] is the correct answer to “The eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ 0&2 \end{array}} \right]\] are written in the form \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{a}} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{b}} \end{array}} \right].\] What is a + b =”. This question is standard in competitive examination syllabi for Linear Algebra.