# Linear Algebra

## Theorems

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

## Matrices

### Basic Problems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Inverse & Rank of a Matrix

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

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

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

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

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

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

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

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

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

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

## Inner Products

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

## Vector Algebra

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Vector Product

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

### Jordan

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