If A= [ ccc 2 & -3 & 5 3 & 2 & -4 1 & 1 & -2 ] find A^ -1 , using… - Vidyadip

Question

If $A=\left[\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right]$ find $A^{-1}$, using $A^{-1}$ solve the system of equations
$2 x-3 y+5 z=11$
$3 x+2 y-4 z=-5$
$x+y-2 z=-3$

✓

Answer

$A = \left[ {\begin{array}{*{20}{c}} 2&{ - 3}&5 \\ 3&2&{ - 4} \\ 1&1&{ - 2} \end{array}} \right]$
$\therefore |A| = 2(-4 + 4) + 3(-6 +4)+ 5(3 - 2) = 0 -6 +5 = -1$
Now$, A_{11} = 0, A_{12}= 2, A_{13}= 1$
$A_{21}= -1, A_{22} = -9, A_{23}= -5$
$A_{31}= 2, A_{32}= 23,A_{33} = 13$
$\therefore {A^{ - 1}} = \frac{1}{{|A|}}(adjA) = - \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&2 \\ 2&{ - 9}&{23} \\ 1&{ - 5}&{13} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&1&{ - 2} \\ { - 2}&9&{ - 23} \\ { - 1}&5&{ - 13} \end{array}} \right]....(1)$
Now, the given system of equations can be written in the form of $AX = B,$ where
$A = \left[ {\begin{array}{*{20}{c}} 2&{ - 3}&5 \\ 3&2&{ - 4} \\ 1&1&{ - 2} \end{array}} \right]\;,X = \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right]\;and\;B = \left[ {\begin{array}{*{20}{c}} {11} \\ { - 5} \\ { - 3} \end{array}} \right]$
The solution of the system of equations is given by $X = A^{-1}B$
$\Rightarrow \left[ {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&1&{ - 2} \\ { - 2}&9&{ - 23} \\ { - 1}&5&{ - 13} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {11} \\ { - 5} \\ { - 3} \end{array}} \right] \ [$Using $(1) ]$
$ = \left[ {\begin{array}{*{20}{c}} {0 - 5 + 6} \\ { - 22 - 45 + 69} \\ { - 11 - 25 + 39} \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right]$
Hence$, x = 1, y = 2$ 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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the values of a and b such that the function f defined by $\text{f(x)}=\begin{cases}\frac{\text{x}-4}{|\text{x}-4|}+\text{a},&\text{if x}<4\\\text{a+}\text{b},&\text{if x}=4\\\frac{\text{x}-4}{|\text{x}-4|}+\text{b},&\text{if x}>4\end{cases}$ is a continuous function at x = 4.

Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$

In order to supplement daily diet, a person wishes to take $X$ and $Y$ tablets. The contents $($in milligrams per tablet$)$ of iron, calcium and vitamins in $X$ and $Y$ are given as below:

Tablets	Iron	Calcium	Vitamin
$X$	$6$	$3$	$2$
$Y$	$2$	$3$	$4$

The person needs to supplement at least $18$ milligrams of iron, $21$ milligrams of calcium and $16$ milligrams of vitamins. The price of each tablet of $X$ and $Y$ is $₹ 2$ and $₹1$ respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an $LPP$ and solve graphically.

A diet is to contain at least $80$ units of vitamin $A$ and $100$ units of minerals. Two foods $F_1$ and $F_2$ are available. Food $F_1$ costs $Rs. 4$ per unit and $F_2$ costs $Rs. 6$ per unit one unit of food $F_1$ contains $3$ units of vitamin $A$ and $4$ units of minerals. One unit of food $F_2$ contains $6$ units of vitamin $A$ and $3$ units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements.

Using matrices, solve the following system of equations:
$x + 2y + z = 7.$
$x + 3z = 11.$
$3x - 3y = 1.$

Using differentials, find the approximate values of the following:
$\sqrt{37}$

What are the values of $'a\ '$ for which $f(x) = a^x$ is decreasing on $R$?

Find the shortest distance between the lines
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and, $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$

An urn contains 5 red and 2 blcak balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, then find the mean and variance of X.

A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing,

Two red balls,
Two black balls,
First red and second black ball.