Determinant

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #91

Prove that $ \begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & y+x \end{vmatrix} = 4xyz $

Question No: #92

Show that (without expanding) $ \begin{vmatrix} 1 & x^2+yz & x^3 \\ 1 & y^2+zx & y^3 \\ 1 & z^2+xy & z^3 \end{vmatrix} = -(x-y) (y-z) (z-x) (x^2+y^2+z^2)$

Question No: #93

If `c` is any constant, then prove that $ \begin{vmatrix} x & x^2 & 1+cx^3 \\ y & y^2 & 1+cy^3 \\ z & z^2 & 1+cz^3 \end{vmatrix} = (1+cxyz)(x-y)(y-z)(z-x) $

Question No: #94

If `x\ne\y\nez` and $ \begin{vmatrix} x & x^2 & 1+x^3 \\ y & y^2 & 1+y^3 \\ z & z^2 & 1+z^3 \end{vmatrix} =0 $, then show that `1+xyz=0`

Question No: #95

Prove that $ \begin{vmatrix} \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{vmatrix} = (\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)(\alpha+\beta+\gamma)$

Question No: #96

Prove that $ \begin{vmatrix} a^2+\lambda & ab & ac \\ ab & b^2+\lambda & bc \\ ac & bc & c^2+\lambda \end{vmatrix} = \lambda^2(a^2+b^2+c^2+\lambda) $

Question No: #97

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

Question No: #98

Show that $ \begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ca & bc & -c^2 \end{vmatrix} = 4a^2b^2c^2 $

Question No: #99

Prove that $ \begin{vmatrix} -bc & b^2+bc & c^2+bc \\ a^2+ac & -ac & c^2+ac \\ a^2+ab & b^2+ab & -ab \end{vmatrix} = (ab+bc+ca)^3 $

Question No: #100

Prove that $ \begin{vmatrix} 1+a^2-b^2 & 2ab & -2b \\ 2ab & 1-a^2+b^2 & 2a \\ 2b & -2a & 1-a^2-b^2 \end{vmatrix} = (1+a^2+b^2)^3 $