Determinant

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

Click on each question to see answer & solution.

Question No: #111

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

Question No: #112

Prove that: $ \begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & b+a & c \end{vmatrix} = (a+b+c)(a^2+b^2+c^2) $

Question No: #113

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

Question No: #114

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

Question No: #115

Prove that $ \begin{vmatrix} -1 & b & c \\ a & -1 & c \\ a & b & -1 \end{vmatrix} = (a+1)(b+1)(c+1)(\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}-1) $

Question No: #116

Prove that $ \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix} = xy+yz+zx+xyz = xyz\ (1+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) $

Question No: #117

Without expanding prove that: $ \begin{vmatrix} x & b & c \\ a & y & c \\ a & b & z \end{vmatrix} $`=(x-a)(y-b)(z-c)(\frac{x}{x-a}+\frac{y}{y-b}+\frac{z}{z-c}-2)`

Question No: #118

If `a\ne\p`, `b\ne\q`, `c\ne\r` and $ \begin{vmatrix} p & b & c \\ a & q & c \\ a & b & r \end{vmatrix} = 0 $, then show that `\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}=2`.

Question No: #119

If $ \begin{vmatrix} p & q-y & r-z \\ p-x & q & r-z \\ p-x & q-y & r \end{vmatrix} = 0 $ , then show that `p/x+q/y+r/z = 2`

Question No: #120

Prove that $ \begin{vmatrix} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{vmatrix} = 4abc $