Left| {,begin{array}{*{20}{c}}{1 + i}&{1 - i}&i{1

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

medium

$\left| {\,\begin{array}{*{20}{c}}{1 + i}&{1 - i}&i\\{1 - i}&i&{1 + i}\\i&{1 + i}&{1 - i}\end{array}\,} \right| = $

$ - 4 - 7i$

$4 + 7i$

$3 + 7i$

$7 + 4i$

Solution

(b) $\Delta = (2 + i)\,\left| {\,\begin{array}{*{20}{c}}1&1&i\\1&{1 + 2i}&{1 + i}\\1&2&{1 – i}\end{array}\,} \right|\,$

=$(2 + i)$$\left| {\,\begin{array}{*{20}{c}}0&{ – 2i}&{ – 1}\\0&{ – 1 + 2i}&{2i}\\1&2&{1 – i}\end{array}\,} \right|$ by $\begin{array}{l}{R_1} \to {R_1} – {R_2}\\{R_2} \to {R_2} – {R_3}\end{array}$

= $(2 + i)\,\,\{ – 4{i^2} + ( – 1 + 2i)\} = (2 + i)\,(4 – 1 + 2i)$

= $(2 + i)\,(3 + 2i) = 4 + 7i$.

Standard 12

Mathematics

$\left| {\,\begin{array}{*{20}{c}}0&{p – q}&{p – r}\\{q – p}&0&{q – r}\\{r – p}&{r – q}&0\end{array}\,} \right| = $

easy

If $a \ne p,b \ne q,c \ne r$ and $\left| {\,\begin{array}{*{20}{c}}p&b&c\\{p + a}&{q + b}&{2c}\\a&b&r\end{array}\,} \right|$ =$ 0$, then $\frac{p}{{p – a}} + \frac{q}{{q – b}} + \frac{r}{{r – c}} = $

hard

The number of solutions of equations $x + y – z = 0$, $3x – y – z = 0, \,x – 3y + z = 0$ is

medium

The system of linear equation $x + y + z = 2, 2x + 3y + 2z = 5$, $2x + 3y + (a^2 -1)\,z = a + 1$ then

hard

(JEE MAIN-2019)

The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ – x}&1&0\\1&{ – x}&1\\0&1&{ – x}\end{array}\,} \right| = 0 $ is equal to