Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
Solve the system of linear equation, using matrix method $x - y + 2z = 7; 3x + 4y - 5z = -5; 2x - y + 3z = 12$
✓

Answer

Matrix form of given equations is $AX = B \Rightarrow \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&4&{ - 5} \\ 2&{ - 1}&3 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 7 \\ { - 5} \\ {12} \end{array}} \right]$
Here $A = \left[ {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&4&{ - 5} \\ 2&{ - 1}&3 \end{array}} \right],X = \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}} 7 \\ { - 5} \\ {12} \end{array}} \right]$
$\therefore \left| A \right| = \left| {\begin{array}{*{20}{c}} 1&{ - 1}&2 \\ 3&4&{ - 5} \\ 2&{ - 1}&3 \end{array}} \right|$
$ =1(12-5)-(-1)(9+10)+2(-3-8) $
$ =7+19-22=4 \neq 0 $
$ A_{11}=7, A_{12}=-19, A_{13}=-11 $
$ A_{21}=1, A_{22}=-1, A_{23}=-1 $
$ A_{31}=-3, A_{32}=11, A_{33}=7$
adj A=$\left[ {\begin{array}{*{20}{c}} 7&1&{ - 3} \\ { - 19}&{ - 1}&{11} \\ { - 11}&{ - 1}&7 \end{array}} \right]$
Therefore, solution is unique and $X = {A^{ - 1}}B = \frac{1}{{\left| A \right|}}\left( {adj.A} \right)B$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \frac{1}{4}\left[ {\begin{array}{*{20}{c}} 7&1&{ - 3} \\ { - 19}&{ - 1}&{11} \\ { - 11}&{ - 1}&7 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 7 \\ { - 5} \\ {12} \end{array}} \right]$
$= \frac{1}{4}\left[ {\begin{array}{*{20}{c}} {49 - 5 - 36} \\ { - 133 + 5 + 132} \\ { - 77 + 5 + 84} \end{array}} \right]$
$= \frac{1}{4}\left[ {\begin{array}{*{20}{c}} 8 \\ 4 \\ {12} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 1 \\ 3 \end{array}} \right]$
Therefore, $x = 2, y = 1$ and $z = 3$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Determinants→Determinants — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differentiate the following functions with respect to x:
$\text{e}^{\text{ax}}\sec\text{x}\tan2\text{x}$
→
Find the vector equation of the plane through the line of intersection of the planes $x + y + z = 1$ and $2x + 3y + 4z = 5$ which is perpendicular to the plane $x - y + z = 0$.
→
Show that among all positive number x and y with $x^2 + y^2 = r^2$, the sum x + y is largest when x = y =$\sqrt{2}.$
→
Evaluate $\int\limits_0^{3}(2x^2 + 3x + 5)dx$ as limit of a sum.
→
find the area of the region included between the parabola $y^2 = x$ and the line $x + y = 2$.
→
Differentiate $\tan^{-1}\Big(\frac{\sqrt{1 + \text{x}^{2}}-1}{\text{x}}\Big)$with respect to $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^{2}}\Big),$ when $\text{x}\neq0.$
→
Find the equation of a plane which passes through the point (3, 2, 0) and contains the line $\frac{\text{x}-3}{1}=\frac{\text{y}-6}{5}=\frac{\text{z}-4}{4}$
→
A dealer in a rural area wishes to purchase some sewing machines. He has only to invest and has space for at most 20 items. An electronic machine costs him ₹ 3,600 and a manually operated machine costs ₹ 2,400. He can sell an electronic machine at a profit of ₹ 220 and a manually operated machine at a profit of ₹ 180. Assuming that he can sell all the machines that he buys, how should he invest his money in order to maximise his profit? Make it as a LPP and solve it graphically.
→
Differentiate the following functions with respect to x:
$\sqrt{\frac{\text{a}^2-\text{x}^2}{\text{a}^2+\text{x}^2}}$
→
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\frac{\text{e}^\text{t}+\text{e}^{-\text{t}}}{2}\text{ and y}=\frac{\text{e}^\text{t}-\text{e}^\text{-t}}{2}$
→