Question
Find the inverse of the following matrices by the adjoint method : $\left[\begin{array}{rr}-1 & 5 \\ -3 & 2\end{array}\right]$
✓
Answer
Let $A=\left[\begin{array}{rr}-1 & 5 \\ -3 & 2\end{array}\right] $
$\therefore|A|=\left|\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right|=-2+15=13 \neq 0$
$\therefore A^{-1} $ exists.
First we have to find the co $-$ factor matrix
$= [A_{ij}]_{2\times 2}$, where $A_{ij} = (-1)^{i+j}M_{ij}$
Now,$ A_{11} = (-1)^{1+1}M_{11} = 2$
$A_{12} = (-1)^{1+2}M_{12} = -(-3) = 3$
$A_{21} = (-1)^{2+1}M_{21} = -5$
$A_{22} = (-1)^{2+2}M_{22} = -1$
Hence, the co $-$ factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}2 & 3 \\ -5 & -1\end{array}\right] $
$ \therefore \operatorname{adj} A=\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
$ \therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{13}\left[\begin{array}{cc}2 & -5 \\ 3 & -1\end{array}\right]$
$\therefore|A|=\left|\begin{array}{ll}-1 & 5 \\ -3 & 2\end{array}\right|=-2+15=13 \neq 0$
$\therefore A^{-1} $ exists.
First we have to find the co $-$ factor matrix
$= [A_{ij}]_{2\times 2}$, where $A_{ij} = (-1)^{i+j}M_{ij}$
Now,$ A_{11} = (-1)^{1+1}M_{11} = 2$
$A_{12} = (-1)^{1+2}M_{12} = -(-3) = 3$
$A_{21} = (-1)^{2+1}M_{21} = -5$
$A_{22} = (-1)^{2+2}M_{22} = -1$
Hence, the co $-$ factor matrix
$=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right]=\left[\begin{array}{cc}2 & 3 \\ -5 & -1\end{array}\right] $
$ \therefore \operatorname{adj} A=\left[\begin{array}{ll}2 & -5 \\ 3 & -1\end{array}\right]$
$ \therefore A^{-1}=\frac{1}{|A|}(\operatorname{adj} A)=\frac{1}{13}\left[\begin{array}{cc}2 & -5 \\ 3 & -1\end{array}\right]$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Show the equation $2 x^2+x y-y^2+x+4 y-3=0$ represents a pair of lines. Also find the acute angle between them.
→$a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13} = |A|$
→Find the particular solution of the following differential equations:
→$\left( x +2 y ^2\right) \frac{d y}{d x}= y$, when $x =2, y =1$
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E(X).
In ∆ABC, if a cos A = b cos B then prove that the triangle is right angled or an isosceles traingle.
Evaluate : $\int_0^\pi \frac{x \sin x}{1+\sin x} d x$
→Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Find the area of the region lying between the parabolas $4y^2 = 9x$ and $3x^2 = 16y$
→Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y – 11 = 0
The following is the c.d.f. of a r.v. X:
Find
(i) p.m.f. of X
(ii) P( -1 ≤ X ≤ 2)
(iii) P(X ≤ X > 0).