# Linear Algebra

## Theorems

solution Find the eigenvalues of the matrix ${\begin{bmatrix}5&2\\3&6\\\end{bmatrix}}$
solution Define the adjoint of a matrix.
solution Define a unitary matrix.
solution Show that $(A^{*})^{*}=A\,$
solution Show that $(AB^{*})^{*}=BA^{*}\,$
solution Show that $AA^{*}\,$ is self-adjoint.
solution Show that the identity matrix $I\,$ is self-adjoint.
solution Show that the zero matrix ${\mathcal {O}}\,$ is self-adjoint.
solution Show that $(\alpha A+\beta B)^{*}=(\overline {\alpha }A^{*}+\overline {\beta }B^{*})\,$
solution Let $A\,$ be an $n\times n$ matrix such that $A^{2}=A\,$, and let $I\,$ be the $n\times n$ identity matrix. Prove that ${{\rm {rank}}}(A)+{{\rm {rank}}}(A-I)=n\,$.
solution Let $A\,$ be an $m\times n$ matrix. Prove that ${{\rm {Null}}}(A)^{\perp }={{\rm {Col}}}(A^{T})$.

## Matrices

### Basic Problems

solution If $A={\begin{bmatrix}0&1&2\\2&3&4\end{bmatrix}}\,$ and $B={\begin{bmatrix}1&0&0\\2&-3&1\end{bmatrix}}\,$ evaluate $2A+3B\,$

solution Find $x\,$ such that ${\begin{bmatrix}1&x&1\end{bmatrix}}{\begin{bmatrix}1&3&2\\2&5&1\\15&3&2\end{bmatrix}}{\begin{bmatrix}1\\2\\x\end{bmatrix}}=0\,$.

solution If $A={\begin{bmatrix}2&3\\-1&2\end{bmatrix}}\,$ Show that $A^{2}-4A+7I=O\,$

Solution If w is cube root of unity,show that ${{\begin{bmatrix}1&w&w^{2}\\w&w^{2}&1\\w^{2}&1&w\end{bmatrix}}+{\begin{bmatrix}w&w^{2}&1\\w^{2}&1&w\\w&w^{2}&1\end{bmatrix}}}{\begin{bmatrix}1\\w\\w^{2}\end{bmatrix}}={\begin{bmatrix}0\\0\\0\end{bmatrix}}\,$

solution If $A={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}\,$ for all integral values of n,show that $A^{n}={\begin{bmatrix}\cos n\theta &\sin n\theta \\-\sin n\theta &\cos n\theta \end{bmatrix}}\,$

solution Find the value of determinant of the matrix ${\begin{bmatrix}29&26&22\\25&31&27\\63&54&46\end{bmatrix}}\,$

solution Show that ${\begin{vmatrix}bc&b+c&1\\ca&c+a&1\\ab&a+b&1\end{vmatrix}}=(a-b)(b-c)(c-a)\,$

solution Prove that the determinant of the matrix ${\begin{bmatrix}y+z&z&y\\z&z+x&x\\y&x&x+y\end{bmatrix}}=4xyz\,$

solution Show that ${\begin{vmatrix}a+b&b+c&c+a\\b+c&c+a&a+b\\c+a&a+b&b+c\end{vmatrix}}=2{\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}}\,$

solution Without expanding the determinant of the matrix $A={\begin{bmatrix}0&p-q&p-r\\q-p&0&q-r\\r-p&r-q&0\end{bmatrix}}\,$ prove that $|A|=0\,$

solution Prove that ${\begin{vmatrix}-2a&a+b&c+a\\b+a&-2b&b+c\\c+a&c+b&-2c\end{vmatrix}}=4(a+b)(b+c)(c+a)\,$

solution If a,b,c are distinct, $abc\not \equiv 0\,$ and ${\begin{vmatrix}a&a^{3}&a^{4}-1\\b&b^{3}&b^{4}-1\\c&c^{3}&c^{4}-1\end{vmatrix}}=0\,$ then prove that $abc(ab+bc+ca)=a+b+c\,$

solution Show that ${\begin{vmatrix}a&a+b&a+b+c\\2a3a+2b&4a+3b+2c\\3a&6a+3b&10a+6b+3c\end{vmatrix}}=a^{3}\,$

solution Prove that ${\begin{vmatrix}a^{2}&bc&ac+c^{2}\\a^{2}+ab&b^{2}&ac\\ab&b^{2}+bc&c^{2}\end{vmatrix}}=4a^{2}b^{2}c^{2}\,$

solution Without expanding the determinant prove that ${\begin{vmatrix}1&bc&b+c\\a&ca&c+a\\1&ab&a+b\end{vmatrix}}={\begin{vmatrix}1&a&a^{2}\\1&b&b^{2}\\1&c&c^{2}\end{vmatrix}}\,$

solution Solve for x given ${\begin{vmatrix}x-2&2x-3&3x-4\\x-4&2x-9&3x-16\\x-8&2x-27&3x-64\end{vmatrix}}=0\,$

solution Show that the determinant of the matrix ${\begin{bmatrix}\cos(\theta +\alpha )&\sin(\theta +\alpha )&1\\\cos(\theta +\beta )&\sin(\theta +\beta )&1\\\cos(\theta +\alpha )&\sin(\theta +\alpha )&1\end{bmatrix}}\,$ is independent of theta.

solution Show that ${\begin{vmatrix}b+c&a-b&a\\c+a&b-c&b\\a+b&c-a&c\end{vmatrix}}=3abc-a^{3}-b^{3}-c^{3}\,$

### Inverse & Rank of a Matrix

solution If A,B are invertible matrices of the same order,then $(AB)^{{-1}}=B^{{-1}}A^{{-1}}\,$

Solution Compute the adjoint of the matrix $A={\begin{bmatrix}1&2&2\\2&3&0\\0&1&2\end{bmatrix}}\,$

solution Find the adjoint and inverse of $A={\begin{bmatrix}2&3&4\\4&3&1\\1&2&4\end{bmatrix}}\,$

solution Determine the rank of $A={\begin{bmatrix}4&2&3\\8&4&6\\-2&-1&-1.5\end{bmatrix}}\,$

solutionFind the rank of A,rank of B $A={\begin{bmatrix}1&5&4\\0&3&2\\2&3&10\end{bmatrix}}\,$, $B={\begin{bmatrix}1&1&1\\2&2&2\\3&3&3\end{bmatrix}}\,$

solution Determine the values of b such that the rank of the matrix A is 3. $A={\begin{bmatrix}1&1&-1&0\\4&4&-3&1\\b&2&2&2\\9&9&b&3\end{bmatrix}}\,$

solution Find the non-singular matrices P and Q such that the normal form of A is PAQ where $A={\begin{bmatrix}1&3&6&-1\\1&4&5&1\\1&5&4&3\end{bmatrix}}\,$. Hence find its rank.

solution Find P and Q such that the normal form of $A={\begin{bmatrix}1&-1&-1\\1&1&1\\3&1&1\end{bmatrix}}\,$ is PAQ. Hence the find the rank.

solution A=${\begin{bmatrix}1&5&4\\0&3&2\\2&3&10\end{bmatrix}}\,$, B=${\begin{bmatrix}1&1&1\\2&2&2\\3&3&3\end{bmatrix}}\,$. Find the rank of $A+B\,$ and $AB\,$.

solution Solve by Cramer's rule $x+y+z=11,2x-6y-z=0,3x+4y+2z=0\,$

## Inner Products

solution Define an inner product.
solution Show that $\langle x,\alpha y\rangle =\overline {\alpha }\langle x,y\rangle$
solution Show that $\langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle$
solution Show that $\langle x,y\rangle +\langle y,x\rangle =2{\mbox{ Re }}\langle x,y\rangle$
solution Show that $\langle x,y\rangle -\langle y,x\rangle =2i{\mbox{ Im }}\langle x,y\rangle$
solution Show that $\langle \alpha x,\beta y\rangle =\alpha \overline {\beta }\langle x,y\rangle$
solution Show that $\langle \alpha x,\alpha y\rangle =|\alpha |^{2}\langle x,y\rangle$
solution Show that $\langle -x,-y\rangle =\langle x,y\rangle$
solution Show that $\langle x,{\vec {0}}\rangle =0$
solution Show that $\langle {\vec {0}},y\rangle =0$
solution Show that $\langle x,x\rangle$ is always real.

## Vector Algebra

solution If $AB,BE,CF\,$ are the medians of a triangle,then prove that $\quad {\bar {AB}}+{\bar {BE}}+{\bar {CF}}={\bar {O}}\,$

solution If G is the centroid of the triangle ABC,prove that ${\bar {GA}}+{\bar {GB}}+{\bar {GC}}={\bar {O}}\,$ where $A,B,C\,$ are the vertices of the triangle ABC and ${\bar {O}}\,$ is the point vector

solution The position vectors of A and B are $2{\bar {i}}+{\bar {j}}-{\bar {k}},{\bar {i}}+2{\bar {j}}+3{\bar {k}}\,$ respectively.Find the position vector of the point which divides the line segment AB in the ration 2:3.

solution If ${\bar {a}}=2{\bar {i}}+{\bar {k}},{\bar {b}}=3{\bar {i}}+4{\bar {k}},{\bar {c}}=8{\bar {i}}+9{\bar {k}}\,$ then express ${\bar {c}}\,$ as a linear combination of ${\bar {a}}\,$ and ${\bar {b}}\,$.

solution If ${\bar {OA}}={\bar {i}}+{\bar {j}}+{\bar {k}},{\bar {AB}}=3{\bar {i}}+2{\bar {j}}+{\bar {k}},{\bar {BC}}={\bar {i}}+2{\bar {j}}-2{\bar {k}},{\bar {CD}}=2{\bar {i}}+{\bar {j}}+3{\bar {k}}\,$,then find the position vector of D.

solution If ${\bar {a}}\,$ is the position vector whose point is $(3,-2)\,$.Find the coordinates of a point B such that ${\bar {AB}}={\bar {a}}\,$,the coordinates of A are $(-1,5)\,$

solution Find a vector of magnitude 6units which is parallel to the vector ${\bar {i}}+{\sqrt {3}}{\bar {j}}\,$

solution Find the magnitude of the vector $7{\bar {i}}-3{\bar {j}}+5{\bar {k}}\,$

solution If the position vectors of A and B are $2{\bar {i}}-9{\bar {j}}-4{\bar {k}},6{\bar {i}}-3{\bar {j}}+8{\bar {k}}\,$ respectively,find the unit vector in the direction of AB.

solution If the position vectors of A and B are ${\bar {i}}+3{\bar {j}}-7{\bar {k}},5{\bar {i}}-2{\bar {j}}+4{\bar {k}}\,$ respectively,determine the direction cosines of ${\bar {AB}}\,$

solution In a triangle ABC if $A=2{\bar {i}}+4{\bar {j}}-{\bar {k}},B=4{\bar {i}}+5{\bar {j}}+{\bar {k}},C=3{\bar {i}}+6{\bar {j}}-3{\bar {k}}\,$ and D is the mid point of the side BC, then find the length of AD.

solution Show that the points represented by ${\bar {i}}+2{\bar {j}}+3{\bar {k}},3{\bar {i}}+4{\bar {j}}+7{\bar {k}},-3{\bar {i}}-2{\bar {j}}-5{\bar {k}}\,$ are collinear.

solution Show that the points A,B,C,D with position vectors $6{\bar {i}}-7{\bar {j}},16{\bar {i}}-19{\bar {j}}-4{\bar {k}},3{\bar {j}}-6{\bar {k}},2{\bar {i}}+5{\bar {j}}+10{\bar {k}}\,$ are not coplanar.

solution Prove that three points whose vectors are ${\bar {i}}+2{\bar {j}}+3{\bar {k}},-{\bar {i}}-{\bar {j}}+8{\bar {k}},4{\bar {i}}+4{\bar {j}}+6{\bar {k}}\,$ form an equilateral triangle.

solution Show that the triangle ABC whose vertices are $7{\bar {i}}+10{\bar {k}},-{\bar {i}}+6{\bar {j}}+6{\bar {k}},-4{\bar {i}}+9{\bar {j}}+6{\bar {k}}\,$ is isoscles and right angled.

solution Obtain the point of intersection of the line joining the points ${\bar {i}}-2{\bar {j}}-{\bar {k}},2{\bar {i}}+3{\bar {j}}+{\bar {k}}\,$ with the plane through the points $2{\bar {i}}+{\bar {j}}-3{\bar {k}},4{\bar {i}}-{\bar {j}}+2{\bar {k}}\,$ and $3{\bar {i}}+{\bar {k}}\,$

### Vector Product

solution Show that the points whose position vectors are $2{\bar {i}}-{\bar {j}}+{\bar {k}},{\bar {i}}-3{\bar {j}}-5{\bar {k}},3{\bar {i}}-4{\bar {j}}-4{\bar {k}}\,$ are the vertices of a right angled triangle.

solution Find a vector ${\bar {d}}\,$ which is perpendicular to both ${\bar {a}}=4{\bar {i}}+5{\bar {j}}-{\bar {k}},{\bar {b}}={\bar {i}}-4{\bar {j}}+5{\bar {k}}\,$ and ${\bar {d}}\cdot {\bar {c}}=21\,$ where${\bar {c}}=3{\bar {i}}+{\bar {j}}-{\bar {k}}\,$

solution If ${\bar {a}},{\bar {b}},{\bar {c}}\,$ are mutually perpendicular vectors of equal magnitude,show that ${\bar {a}}+{\bar {b}}+{\bar {c}}\,$ is equally inclined to ${\bar {a}},{\bar {b}},{\bar {c}}\,$

solution Dot products of the vectors ${\bar {i}}+{\bar {j}}-3{\bar {k}},{\bar {i}}+3{\bar {j}}-2{\bar {k}},2{\bar {i}}+{\bar {j}}+4{\bar {k}}\,$ are $0,5,8\,$ 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 $2{\bar {i}}+3{\bar {j}}+6{\bar {k}}\,$ as a normal vector.

solution Find the vector equation of the plane through the point $A(3,-2,1)\,$ and perpendicular to the vector $4{\bar {i}}+7{\bar {j}}-4{\bar {k}}\,$.

solution Find the equation of the plane passing through the point $(3,4,5)\,$ and parallel to the plane ${\bar {r}}\cdot (2{\bar {i}}+3{\bar {j}}-{\bar {k}})=6\,$

solution If ${\bar {a}}=2{\bar {i}}-3{\bar {j}}+5{\bar {k}},{\bar {b}}=-{\bar {i}}-4{\bar {j}}+2{\bar {k}}\,$,then write ${\bar {a}}\times {\bar {b}}\,$

solution Determine the unit vector perpendicular to both the vectors $2{\bar {i}}+{\bar {j}}+3{\bar {k}},{\bar {i}}-2{\bar {j}}+{\bar {k}}\,$

solution Find the vector area of a parallelogram whose diagonals are determined by the vectors ${\bar {a}}=3{\bar {i}}+{\bar {j}}-2{\bar {k}},{\bar {b}}={\bar {i}}-3{\bar {j}}+4{\bar {k}}\,$

solution Find a vector of magnitude 3 and which is perpendicular to both the vectors $3{\bar {i}}+{\bar {j}}-4{\bar {k}},6{\bar {i}}+5{\bar {j}}-2{\bar {k}}\,$

solution IF $(1,2,3),(2,5,-1),(-1,1,2)\,$ are the vertices of a triangle, find its area.

solution Find a unit vector perpendiculars to the plane ABC where $A=(3,-1,2),B(1,-1,-3),C(4,-3,1)\,$

solution Let ${\bar {a}}=2{\bar {i}}+3{\bar {j}}+4{\bar {k}},{\bar {b}}={\bar {i}}-2{\bar {j}}+{\bar {k}}\,$.If a vector ${\bar {r}}\,$ satisfies ${\bar {a}}\times {\bar {r}}=3{\bar {b}}\,$ and ${\bar {a}}\cdot {\bar {r}}=2\,$ then find the vector ${\bar {r}}\,$

solution Find the volume of the parallelopiped whose edges are ${\bar {i}}+{\bar {j}}+{\bar {k}},{\bar {i}}-{\bar {j}}+{\bar {k}},{\bar {i}}+{\bar {j}}-{\bar {k}}\,$.

solution Show that the four points having position vectors $6{\bar {i}}-7{\bar {j}},16{\bar {i}}-19{\bar {j}}-4{\bar {k}},3{\bar {i}}-6{\bar {k}},2{\bar {i}}+5{\bar {j}}+10{\bar {k}}\,$ are not coplanar.

solution If the two vectors ${\bar {a}}=2{\bar {i}}+{\bar {j}}+2{\bar {k}},{\bar {b}}=5{\bar {i}}-3{\bar {j}}+{\bar {k}}\,$ are two vectors,find the projection of ${\bar {b}}\,$ on ${\bar {a}}\,$

solution Reduce the equation ${\bar {r}}\cdot (4{\bar {i}}-12{\bar {j}}-3{\bar {k}})+7=0\,$ to normal form and hence find the length of the perpendicular from the origin to the plane.

solution Find the angle between the planes ${\bar {r}}\cdot (2{\bar {i}}-{\bar {j}}+{\bar {k}})=6,{\bar {r}}\cdot ({\bar {i}}+{\bar {j}}+2{\bar {k}})=7\,$

solution Find the value of lambda for which the four points with position vectors $3{\bar {i}}-2{\bar {j}}-{\bar {k}},2{\bar {i}}+3{\bar {j}}-4{\bar {k}},-{\bar {i}}+{\bar {j}}+2{\bar {k}},4{\bar {i}}+5{\bar {j}}+\lambda {\bar {k}}\,$ are coplanar.

solution Find the volume of the tetrahedron with vertices $(1,1,3),(4,3,2),(5,2,7),(6,4,8)\,$

solution Find the vector equation of the line passing through three non-collinear points $-2{\bar {i}}+6{\bar {j}}-6{\bar {k}},-3{\bar {i}}+10{\bar {j}}-9{\bar {k}},-5{\bar {i}}-6{\bar {k}}\,$.Also find its cartesian equation.

solution Find the equation of the plane passing through the points $3{\bar {i}}-5{\bar {j}}-{\bar {k}},-{\bar {i}}+5{\bar {j}}+7{\bar {k}}\,$ and parallel to $3{\bar {i}}-{\bar {j}}+7{\bar {k}}\,$

solution If ${\begin{vmatrix}a&a^{2}&1+a^{3}\\b&b^{2}&1+b^{3}\\c&c^{2}&1+c^{3}\end{vmatrix}}=0\,$ and the vectors ${\bar {A}}=(1,a,a^{2}),{\bar {B}}=(1,b,b^{2}),{\bar {C}}=(1,c,c^{2})\,$ are non-coplanar,then prove that $abc=-1\,$

solution Find the perpendicular distance from the origin to the plane passing through the points $(1,-2,5),(0,-5,-1),(-3,5,0)\,$

### Jordan

solution ${\begin{bmatrix}2&0&-1&1\\1&1&-1&1\\1&-3&2&-1\\0&-3&2&-1\end{bmatrix}}$