Linear Algebra
Contents
Theorems
solution Find the eigenvalues of the matrix
solution Define the adjoint of a matrix.
solution Define a self-adjoint matrix.
solution Define a unitary matrix.
solution Show that
solution Show that
solution Show that is self-adjoint.
solution Show that the identity matrix is self-adjoint.
solution Show that the zero matrix is self-adjoint.
solution Show that
solution Let be an
matrix such that
, and let
be the
identity matrix. Prove that
.
solution Let be an
matrix. Prove that
.
Matrices
Basic Problems
solution If and
evaluate
solution Find such that
.
solution If Show that
Solution If w is cube root of unity,show that
solution If for all integral values of n,show that
solution Find the value of determinant of the matrix
solution Show that
solution Prove that the determinant of the matrix
solution Show that
solution Without expanding the determinant of the matrix prove that
solution Prove that
solution If a,b,c are distinct, and
then prove that
solution Show that
solution Prove that
solution Without expanding the determinant prove that
solution Solve for x given
solution Show that the determinant of the matrix is independent of theta.
solution Show that
Inverse & Rank of a Matrix
solution If A and B are two non-singular matrices of the same type,then adjoint(AB)=(adjoint B)(adjoint A)
solution If A,B are invertible matrices of the same order,then
Solution Compute the adjoint of the matrix
solution Find the adjoint and inverse of
solution Determine the rank of
solutionFind the rank of A,rank of B ,
solution Determine the values of b such that the rank of the matrix A is 3.
solution Find the non-singular matrices P and Q such that the normal form of A is PAQ where . Hence find its rank.
solution Find P and Q such that the normal form of is PAQ. Hence the find the rank.
solution A=, B=
. Find the rank of
and
.
solution Solve by Cramer's rule
Inner Products
solution Define an inner product.
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that
solution Show that is always real.
Vector Algebra
Vector Addition
solution If are the medians of a triangle,then prove that
solution If G is the centroid of the triangle ABC,prove that where
are the vertices of the triangle ABC and
is the point vector
solution The position vectors of A and B are respectively.Find the position vector of the point which divides the line segment AB in the ration 2:3.
solution If then express
as a linear combination of
and
.
solution If ,then find the position vector of D.
solution If is the position vector whose point is
.Find the coordinates of a point B such that
,the coordinates of A are
solution Find a vector of magnitude 6units which is parallel to the vector
solution Find the magnitude of the vector
solution If the position vectors of A and B are respectively,find the unit vector in the direction of AB.
solution If the position vectors of A and B are respectively,determine the direction cosines of
solution In a triangle ABC if and D is the mid point of the side BC, then find the length of AD.
solution Show that the points represented by are collinear.
solution Show that the points A,B,C,D with position vectors are not coplanar.
solution Prove that three points whose vectors are form an equilateral triangle.
solution Show that the triangle ABC whose vertices are is isoscles and right angled.
solution Obtain the point of intersection of the line joining the points with the plane through the points
and
Vector Product
solution Show that the points whose position vectors are are the vertices of a right angled triangle.
solution Find a vector which is perpendicular to both
and
where
solution If are mutually perpendicular vectors of equal magnitude,show that
is equally inclined to
solution Dot products of the vectors are
respectively.Find the vector.
solution Find the vector equation of a plane which is at a distance of 5units from the origin and which has as a normal vector.
solution Find the vector equation of the plane through the point and perpendicular to the vector
.
solution Find the equation of the plane passing through the point and parallel to the plane
solution If ,then write
solution Determine the unit vector perpendicular to both the vectors
solution Find the vector area of a parallelogram whose diagonals are determined by the vectors
solution Find a vector of magnitude 3 and which is perpendicular to both the vectors
solution IF are the vertices of a triangle, find its area.
solution Find a unit vector perpendiculars to the plane ABC where
solution Let .If a vector
satisfies
and
then find the vector
solution Find the volume of the parallelopiped whose edges are .
solution Show that the four points having position vectors are not coplanar.
solution If the two vectors are two vectors,find the projection of
on
solution Reduce the equation to normal form and hence find the length of the perpendicular from the origin to the plane.
solution Find the angle between the planes
solution Find the value of lambda for which the four points with position vectors are coplanar.
solution Find the volume of the tetrahedron with vertices
solution Find the vector equation of the line passing through three non-collinear points .Also find its cartesian equation.
solution Find the equation of the plane passing through the points and parallel to
solution If and the vectors
are non-coplanar,then prove that
solution Find the perpendicular distance from the origin to the plane passing through the points
Jordan