GATE IN
Engineering Mathematics
Linear Algebra
Previous Years Questions

## Marks 1

The figure shows a shape $$ABC$$ and its minor image $${A_1}$$$${B_1}$$$${C_1}$$ across the horizontal axis ($$x$$-axis). The coordinate transformati...
The eigen values of the matrix $$A = \left[ {\matrix{ 1 & { - 1} & 5 \cr 0 & 5 & 6 \cr 0 & { - 6} & 5 \cr } }... If$$V$$is a non-zero vector of dimension$$3 \times 1,$$then the matrix$$\,A = V{V^T}$$has a rank$$=$$________ Let$$A$$be an$$n \times n$$matrix with rank$$r\left( {0 < r < n} \right).$$Then$$AX=0$$has$$p$$independent solutions, where$$p$$is A scalar valued function is defined as$$f\left( x \right){x^T}Ax + {b^T}x + c,$$where$$A$$is a symmetric positive definite matrix with dimension ... For the matrix$$A$$satisfying the equation given below, the eigen values are$$\$\left[ A \right]\left[ {\matrix{ 1 & 2 & 3 \cr 7 &...
The dimension of the null space of the matrix $$\left[ {\matrix{ 0 & 1 & 1 \cr 1 & { - 1} & 0 \cr { - 1} & 0 & { ... Given that$$A = \left[ {\matrix{ { - 5} & { - 3} \cr 2 & 0 \cr } } \right]$$and$${\rm I} = \left[ {\matrix{ 1 & 0 \cr ...
The matrix $$M = \left[ {\matrix{ { - 2} & 2 & { - 3} \cr 2 & 1 & 6 \cr { - 1} & { - 2} & 0 \cr } } \right]$$...
A real $$n \times n$$ matrix $$A$$ $$= \left[ {{a_{ij}}} \right]$$ is defined as follows $$\left\{ {\matrix{ {{a_{ij}} = i,} & {\forall i = j...$$X$$and$$Y$$are non-zero square matrices of size$$n \times n$$. If$$XY = {O_{n \times n}}$$then The eigen values of a$$2 \times 2$$matrix$$X$$are$$-2$$and$$-3$$. The eigen values of matrix$${\left( {X + 1} \right)^{ - 1}}\left( {X + 5{\rm...
Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$ Then the rank of $$A$$ is
Identity which one of the following is an eigen vectors of the matrix $$A = \left[ {\matrix{ 1 & 0 \cr { - 1} & { - 2} \cr } } \ri... Let$$A$$be$$3 \times 3$$matrix with rank$$2.$$Then$$AX=O$$has The necessary condition to diagonalize a matrix is that The rank of matrix$$A = \left[ {\matrix{ 1 & 2 & 3 \cr 3 & 4 & 5 \cr 4 & 6 & 8 \cr } } \right]$$is ## Marks 2 Consider the matrix$$A = \left( {\matrix{ 2 & 1 & 1 \cr 2 & 3 & 4 \cr { - 1} & { - 1} & { - 2} \cr } } \righ...
One pair of eigenvectors corresponding to the two eigen values of the matrix $$\left[ {\matrix{ 0 & { - 1} \cr 1 & {0 - } \cr } } ... Let$$A$$be$$n \times n$$real matrix such that$${A^2} = {\rm I}$$and$$Y$$be an$$n$$-diamensional vector. Then the linear system of equations ... A system of linear simultaneous equations is given as$$AX=b$$where$$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 &...
A system of linear simultaneous equations is given as $$AX=b$$ where $$A = \left[ {\matrix{ 1 & 0 & 1 & 0 \cr 0 & 1 & 0 &... For a given$$2x2$$matrix$$A,$$it is observved that$$A\left[ {\matrix{ 1 \cr { - 1} \cr } } \right] = - 1\left[ {\matrix{ 1 \cr ...
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