GATE CE
Engineering Mathematics
Linear Algebra
Previous Years Questions

## Marks 1

Consider the following simultaneous equations (with $${c_1}$$ and $${c_2}$$ being constants): $$3{x_1} + 2{x_2} = {c_1}$$$$$4{x_1} + {x_2} = {c_2}... The matrix$$P$$is the inverse of a matrix$$Q.$$If$${\rm I}$$denotes the identity matrix, which one of the following options is correct? If the entries in each column of a square matrix$$M$$add up to$$1$$, then an eigenvalue of$$M$$is Let$$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$with$$n \ge 3$$and$${{a_{ij}} = i.j.}$$The rank of$$A$$is : For what value of$$'p'$$the following set of equations will have no solutions?$$$2x+3y=5$3x+py=10$$Given the matrices$$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$and$$K = ... The sum of Eigen values of the matrix, $$\left[ M \right]$$ is where $$\left[ M \right] = \left[ {\matrix{ {215} & {650} & {795} \cr ... The determinant of matrix$$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr ... The rank of the matrix $$\left[ {\matrix{ 6 & 0 & 4 & 4 \cr { - 2} & {14} & 8 & {18} \cr {14} & { - 14} &... The eigen values of matrix$$\left[ {\matrix{ 9 & 5 \cr 5 & 8 \cr } } \right]$$are In the solution of the following set of linear equations by Gauss-elimination using partial pivoting$$$5x+y+2z=34,$4y-3z=12$$and$$$10x-2y+z...
A square matrix $$B$$ is symmetric if ____
The product of matrices $${\left( {PQ} \right)^{ - 1}}P$$ is
The eigenvalues of the matrix $$\left[ P \right] = \left[ {\matrix{ 4 & 5 \cr 2 & { - 5} \cr } } \right]$$ are
The following system of equations $$x+y+z=3,$$$$$x+2y+3z=4,$$$ $$x+4y+kz=6$$$will not have a unique solution for $$k$$ equal to Solution for the system defined by the set of equations $$4y+3z=8, 2x-z=2$$ & $$3x+2y=5$$ is Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$2{x_1} - {x_2} + 3{x_3} = 1$$$ $$3{x_1} + 2{... Consider the matrices$$\,{X_{4x3,}}\,\,{Y_{4x3}}\,\,{P_{2x3}}.$$The order of$$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}...
Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda$$ is a scalar. Let $$\left( {{\lambda _i},\,\,... Real matrices$$\,\,{\left[ A \right]_{3x1,}}\,\,{\left[ B \right]_{3x3,}}\,\,{\left[ C \right]_{3x5,}}\,\,{\left[ D \right]_{5x3,}}$$... The eigen values of the matrix$$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$are Given matrix$$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 ...
Eigen values of the following matrix are $$\left[ {\matrix{ { - 1} & 4 \cr 4 & { - 1} \cr } } \right]$$
The determinant of the following matrix $$\left[ {\matrix{ 5 & 3 & 2 \cr 1 & 2 & 6 \cr 3 & 5 & {10} \cr } } \... The eigen values of the matrix$$\left[ {\matrix{ 5 & 3 \cr 2 & 9 \cr } } \right]$$are The product$$\left[ P \right]\,\,{\left[ Q \right]^T}$$of the following two matrices$$\left[ P \right]\,$$and$$\left[ Q \right]\,$$where$$\left...
Consider the following two statements. $$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A.... If$$A,B,C$$are square matrices of the same order then$${\left( {ABC} \right)^{ - 1}}$$is equal be If$$A$$is any$$nxn$$matrix and$$k$$is a scalar then$$\left| {kA} \right| = \alpha \left| A \right|$$where$$\alpha $$is The number of terms in the expansion of general determinant of order$$n$$is The equation$$\left[ {\matrix{ 2 & 1 & 1 \cr 1 & 1 & { - 1} \cr y & {{x^2}} & x \cr } } \right] = 0$$repres... In matrix algebra$$AS=AT$$($$A,S,T,$$are matrices of appropriate order) implies$$S=T$$only if If$$A$$is a real square matrix then$$A{A^T}$$is The real symmetric matrix$$C$$corresponding to the quadratic form$$Q = 4{x_1}{x_2} - 5{x_2}{x_2}$$is Obtain the eigen values and eigen vectors of$$A = \left[ {\matrix{ 8 & -4 \cr 2 & { 2 } \cr } } \right].$$If$$A$$and$$B$$are two matrices and$$AB$$exists then$$BA$$exists, Inverse of matrix$$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right]$$is If the determinant of the matrix$$\left[ {\matrix{ 1 & 3 & 2 \cr 0 & 5 & { - 6} \cr 2 & 7 & 8 \cr } } \right...

## Marks 2

If $$A = \left[ {\matrix{ 1 & 5 \cr 6 & 2 \cr } } \right]\,\,and\,\,B = \left[ {\matrix{ 3 & 7 \cr 8 & 4 \cr }... Consider the matrix$$\left[ {\matrix{ 5 & { - 1} \cr 4 & 1 \cr } } \right].$$Which one of the following statements is TRUE for t... Consider the following linear system$$$x+2y-3z=a$2x+3y+3z=b$5x+9y-6z=c$$This system is consistent if$$a,b$$and$$c$$satisfy the equ... The two Eigen Values of the matrix$$\left[ {\matrix{ 2 & 1 \cr 1 & p \cr } } \right]$$have a ratio of$$3:1$$for$$p=2.$$What ... The smallest and largest Eigen values of the following matrix are :$$\left[ {\matrix{ 3 & { - 2} & 2 \cr 4 & { - 4} & 6 \cr... What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix ... The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is The inverse of $$2 \times 2$$ matrix $$\left[ {\matrix{ 1 & 2 \cr 5 & 7 \cr } } \right]$$ is For what values of $$\alpha$$ and $$\beta$$ the following simultaneous equations have an infinite number of solutions $$x+y+z=5,$$$ $$x+3y+3z=9,... The minimum and maximum eigen values of matrix$$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr }...
For a given matrix A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } \right...
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