GATE ECE
Engineering Mathematics
Linear Algebra
Previous Years Questions

## Marks 1

Consider a system of linear equations Ax = b, where $$A = \left[ {\matrix{ 1 \hfill & { - \sqrt 2 } \hfill & 3 \hfill \cr { - 1} \hfill & {\sq... Let$$\alpha$$,$$\beta$$two non-zero real numbers and v1, v2 be two non-zero real vectors of size 3$$\times$$1. Suppose that v1 and v2 satisfy$$v...
Let M be a real 4 $$\times$$ 4 matrix. Consider the following statements : S1: M has 4 linearly independent eigenvectors. S2: M has 4 distinct eige...
Consider matrix $$A = \left[ {\matrix{ k & {2k} \cr {{k^2} - k} & {{k^2}} \cr } } \right]$$ and vector $$X = \left[ {\matrix{ {... The rank of the matrix$$M = \left[ {\matrix{ 5 & {10} & {10} \cr 1 & 0 & 2 \cr 3 & 6 & 6 \cr } } \right]$$i... Consider the$$5 \times 5$$matrix$$A = \left[ {\matrix{ 1 & 2 & 3 & 4 & 5 \cr 5 & 1 & 2 & 3 & 4 \cr 4 &...
Consider a $$2 \times 2$$ square matrix $$A = \left[ {\matrix{ \sigma & x \cr \omega & \sigma \cr } } \right]$$ Where $$x$$ is ...
The value of $$x$$ for which the matrix $$A = \left[ {\matrix{ 3 & 2 & 4 \cr 9 & 7 & {13} \cr { - 6} & { - 4} & {... Let$${M^4} = {\rm I}$$(where$${\rm I}$$denotes the identity matrix) and$$M \ne {\rm I},\,\,{M^2} \ne {\rm I}$$and$${M^3} \ne {\rm I}$$. Then, ... For$$A = \left[ {\matrix{ 1 & {\tan x} \cr { - \tan x} & 1 \cr } } \right],$$the determinant of$${A^T}\,{A^{ - 1}}$$is The value of$$'x'$$for which all the eigenvalues of the matrix given below are real is$$\left[ {\matrix{ {10} & {5 + j} & 4 \cr x ...
Consider system of linear equations : $$x-2y+3z=-1$$$$$x-3y+4z=1$$$ and $$-2x+4y-6z=k,$$$The value of $$'k'$$ for which the system has infinite... The value of $$'P'$$ such that the vector $$\left[ {\matrix{ 1 \cr 2 \cr 3 \cr } } \right]$$ is an eigenvector of the matrix $$\left... Which one of the following statements is NOT true for a square matrix$$A$$? The system of linear equations$$\left( {\matrix{ 2 & 1 & 3 \cr 3 & 0 & 1 \cr 1 & 2 & 5 \cr } } \right)\left(... The determinant of matrix $$A$$ is $$5$$ and the determinant of matrix $$B$$ is $$40.$$ The determinant of matrix $$AB$$ is _______. The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______. For matrices of same dimension $$M,N$$ and scalar $$c,$$ which one of these properties DOES NOT ALWAYS hold ? Consider the matrix $${J_6} = \left[ {\matrix{ 0 & 0 & 0 & 0 & 0 & 1 \cr 0 & 0 & 0 & 0 & 1 & 0 \cr ...$$A$$real$$\left( {4\,\, \times \,\,4} \right)$$matrix$$A$$satisfies the equation$${A^2} = {\rm I},$$where$${\rm I}$$is the$$\left( {4\,\, \... The minimum eigenvalue of the following matrix is $$\left[ {\matrix{ 3 & 5 & 2 \cr 5 & {12} & 7 \cr 2 & 7 & 5 \c... Let$$A$$be an$$m\,\, \times \,\,n$$matrix and$$B$$an$$n\,\, \times \,\,m$$matrix. It is given that determinant$$\left( {{{\rm I}_m} + AB} \ri... Given that $$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$ and $${\rm I} = \left[ {\matrix{ 1 & 0 \cr ... The system of equations$$x+y+z=6,x+4y+6z=20,x + 4y + \lambda z = \mu $$has no solution for values of$$\lambda $$and$$\mu $$given by The eigen values of a skew-symmetric matrix are All the four entries of$$2x2$$matrix$$P = \left[ {\matrix{ {{p_{11}}} & {{p_{12}}} \cr {{p_{21}}} & {{p_{22}}} \cr }... The system of linear equations $$\left. {\matrix{ {4x + 2y = 7} \cr {2x + y = 6} \cr } } \right\}$$ has For the matrix $$\left[ {\matrix{ 4 & 2 \cr 2 & 4 \cr } } \right].$$ The eigen value corresponding to the eigen vector $$\left[ {\... The rank of the matrix$$\left[ {\matrix{ 1 & 1 & 1 \cr 1 & { - 1} & 0 \cr 1 & 1 & 1 \cr } } \right]$$is The eigen values of the matrix$$\left[ {\matrix{ 2 & { - 1} & 0 & 0 \cr 0 & 3 & 0 & 0 \cr 0 & 0 & { - 2}... The eigen values of the matrix $$A = \left[ {\matrix{ 0 & 1 \cr 1 & 0 \cr } } \right]$$ are The following system of equations $${{x_1} + {x_2} + {x_3} = 3}$$ $${{x_1} - {x_3} = 0}$$ $${{x_1} - {x_2} + {x_3} = 1}$$ has The rank of $$\left( {m \times n} \right)$$ matrix $$\left( {m < n} \right)$$ cannot be more then ## Marks 2 The rank of the matrix $$\left[ {\matrix{ 1 & { - 1} & 0 & 0 & 0 \cr 0 & 0 & 1 & { - 1} & 0 \cr 0 & 1... If the vectors$${e_1} = \left( {1,0,2} \right),\,{e_2} = \left( {0,1,0} \right)$$and$${e_3} = \left( { - 2,0,1} \right)$$form an orthogonal basis ... The matrix$$A = \left[ {\matrix{ a & 0 & 3 & 7 \cr 2 & 5 & 1 & 3 \cr 0 & 0 & 2 & 4 \cr 0 & ... A sequence $$x\left[ n \right]$$ is specified as $$\left[ {\matrix{ {x\left[ n \right]} \cr {x\left[ {n - 1} \right]} \cr } } \right] = ... The eigen values of the following matrix$$\left[ {\matrix{ { - 1} & 3 & 5 \cr { - 3} & { - 1} & 6 \cr 0 & 0 & 3 ... The eigen values and the correspondinng eigen vectors of a $$2 \times 2$$ matrix are given by Eigen value $${\lambda _1} = 8$$ $${\lambda _2} = 4$$ E... Given the matrix $$\left[ {\matrix{ { - 4} & 2 \cr 4 & 3 \cr } } \right],$$ the eigen vector is Given an orthogonal matrix $$A = \left[ {\matrix{ 1 & 1 & 1 & 1 \cr 1 & 1 & { - 1} & { - 1} \cr 1 & { - 1} &a... If$$A = \left[ {\matrix{ 2 & { - 0.1} \cr 0 & 3 \cr } } \right]$$and$${A^{ - 1}} = \left[ {\matrix{ {{\raise0.5ex\hbox{$\scr...
EXAM MAP
Joint Entrance Examination