GATE ME
Engineering Mathematics
Linear Algebra
Previous Years Questions

## Marks 1

The determinant of a $$2 \times 2$$ matrix is $$50.$$ If one eigenvalue of the matrix is $$10,$$ the other eigenvalue is __________.
The product of eigenvalues of the matrix $$P$$ is $$P = \left[ {\matrix{ 2 & 0 & 1 \cr 4 & { - 3} & 3 \cr 0 & 2 &... A real square matrix$$A$$is called skew-symmetric if The condition for which the eigenvalues of the matrix$$A = \left[ {\matrix{ 2 & 1 \cr 1 & k \cr } } \right]$$are positive, is The solution to the system of equations is$$\left[ {\matrix{ 2 & 5 \cr { - 4} & 3 \cr } } \right]\left\{ {\matrix{ x \cr ...
The lowest eigen value of the $$2 \times 2$$ matrix $$\left[ {\matrix{ 4 & 2 \cr 1 & 3 \cr } } \right]$$ is ______.
At least one eigenvalue of a singular matrix is
If any two columns of a determinant $$P = \left| {\matrix{ 4 & 7 & 8 \cr 3 & 1 & 5 \cr 9 & 6 & 2 \cr } } \rig... Which one of the following equations is a correct identity for arbitrary$$3 \times 3$$real matrices$$P,Q$$and$$R$$? Consider a$$3 \times 3$$real symmetric matrix$$S$$such that two of its eigen values are$$a \ne 0,b\,\, \ne 0$$with respective eigen vectors... Given that the determinant of the matrix$$\left[ {\matrix{ 1 & 3 & 0 \cr 2 & 6 & 4 \cr { - 1} & 0 & 2 \cr } ...
Which one of the following describes the relationship among the three vectors, $$\widehat i + \widehat j + \widehat k,\,\,2\widehat i + 3\widehat j +... The eigen values of a symmetric matrix are all For the matrix$$A = \left[ {\matrix{ 5 & 3 \cr 1 & 3 \cr } } \right],$$ONE of the normalized eigen vectors is given as Eigen values of a real symmetric matrix are always Consider the following system of equations$$2{x_1} + {x_2} + {x_3} = 0,\,\,{x_2} - {x_3} = 0$$and$${x_1} + {x_2} = 0.$$This system has One of the eigen vector of the matrix$$A = \left[ {\matrix{ 2 & 2 \cr 1 & 3 \cr } } \right]$$is For a matrix$$\left[ M \right] = \left[ {\matrix{ {{3 \over 5}} & {{4 \over 5}} \cr x & {{3 \over 5}} \cr } } \right].$$The tran... The matrix$$\left[ {\matrix{ 1 & 2 & 4 \cr 3 & 0 & 6 \cr 1 & 1 & P \cr } } \right]$$has one eigen value to ... For what values of 'a' if any will the following system of equations in$$x, y$$and$$z$$have a solution?$$$2x+3y=4,$x+y+z=4,$x+2y-z=a$...
If a square matrix $$A$$ is real and symmetric then the eigen values
The number of linearly independent eigen vectors of $$\left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$$ is
$$A$$ is a $$3 \times 4$$ matrix and $$AX=B$$ is an inconsistent system of equations. The highest possible rank of $$A$$ is
For what value of $$x$$ will the matrix given below become singular? $$\left[ {\matrix{ 8 & x & 0 \cr 4 & 0 & 2 \cr {12} ... The sum of the eigen values of the matrix$$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \r...
For the following set of simultaneous equations $$1.5x - 0.5y + z = 2$$$$$4x + 2y + 3z = 9$$$ $$7x + y + 5z = 10$$$In the Gauss - elimination for a solving system of linear algebraic equations, triangularization leads to The eigen values of $$\left[ {\matrix{ 1 & 1 & 1 \cr 1 & 1 & 1 \cr 1 & 1 & 1 \cr } } \right]$$ are Solve the system $$2x+3y+z=9,$$ $$4x+y=7,$$ $$x-3y-7z=6$$ Among the following, the pair of vectors orthogonal to each other is Find out the eigen value of the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 2 & 3 & 1 \cr 0 & 2 & 4 \cr } } \... ## Marks 2 Consider the matrix$$A = \left[ {\matrix{ {50} & {70} \cr {70} & {80} \cr } } \right]$$whose eigenvectors corresponding to eigen... The number of linear independent eigenvectors of matrix$$A = \left[ {\matrix{ 2 & 1 & 0 \cr 0 & 2 & 0 \cr 0 & 0 &... For a given matrix $$P = \left[ {\matrix{ {4 + 3i} & { - i} \cr i & {4 - 3i} \cr } } \right],$$ where $$i = \sqrt { - 1} ,$$ the... Choose the CORRECT set of functions, which are linearly dependent. $$x+2y+z=4, 2x+y+2z=5, x-y+z=1$$ The system of algebraic equations given above has The eigen vectors of the matrix $$\left[ {\matrix{ 1 & 2 \cr 0 & 2 \cr } } \right]$$ are written in the form $$\left[ {\matrix{ ... Eigen values of a matrix$$S = \left[ {\matrix{ 3 & 2 \cr 2 & 3 \cr } } \right]$$are$$5$$and$$1.$$What are the eigen values o... Multiplication of matrices$$E$$and$$F$$is$$G.$$Matrices$$E$$and$$G$$are$$E = \left[ {\matrix{ {\cos \theta } & { - sin\theta } &... Which one of the following is an eigen vector of the matrix$\$\left[ {\matrix{ 5 & 0 & 0 & 0 \cr 0 & 5 & 0 & 0 \cr ...
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