Question
If $A=\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right], B=\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$ find matrix $C$ such that $5A + 5B + 2C$ is a null matrix.
✓
Answer
Let $C=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
We have $A =\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right], B =\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$
Now $5 A +3 B +2 C =0$
$ \Rightarrow 5\left[\begin{array}{ll} 9 & 1 \\ 7 & 8 \end{array}\right]+3\left[\begin{array}{cc}
1 & 5 \\ 7 & 12 \end{array}\right]+2\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\ 0 & 0\end{array}\right] $
$\Rightarrow\left[\begin{array}{cc} 45 & 5 \\ 35 & 40 \end{array}\right]+\left[\begin{array}{cc} 3 & 15 \\ 21 & 36
\end{array}\right]+\left[\begin{array}{ll} 2 a & 2 b \\ 2 c & 2 d \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\
0 & 0 \end{array}\right]$
$\Rightarrow\left[\begin{array}{cc} 45+3+2 a & 5+15+2 b \\ 35+21+2 c & 40+36+2 d \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] $
$\Rightarrow\left[\begin{array}{ll} 48+2 a & 20+2 b \\ 56+2 c & 76+2 d \end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\ 0 & 0 \end{array}\right] $
$\Rightarrow 48+2 a =0$
$ \Rightarrow 2 a =-48$
$ \Rightarrow a =-24= 20+2 b =0 $
$\Rightarrow 2 b =20 $
$\Rightarrow b =-10 = 56+2 c =0$
$ \Rightarrow 2 c =56 $
$\Rightarrow c =-28 = 76+2 d =0 $
$\Rightarrow 2 d =-76 $
$\Rightarrow d =-38$
Thus $C =\left[\begin{array}{ll} -24 & -10 \\ -28 & -38\end{array}\right] .$
We have $A =\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right], B =\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$
Now $5 A +3 B +2 C =0$
$ \Rightarrow 5\left[\begin{array}{ll} 9 & 1 \\ 7 & 8 \end{array}\right]+3\left[\begin{array}{cc}
1 & 5 \\ 7 & 12 \end{array}\right]+2\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\ 0 & 0\end{array}\right] $
$\Rightarrow\left[\begin{array}{cc} 45 & 5 \\ 35 & 40 \end{array}\right]+\left[\begin{array}{cc} 3 & 15 \\ 21 & 36
\end{array}\right]+\left[\begin{array}{ll} 2 a & 2 b \\ 2 c & 2 d \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\
0 & 0 \end{array}\right]$
$\Rightarrow\left[\begin{array}{cc} 45+3+2 a & 5+15+2 b \\ 35+21+2 c & 40+36+2 d \end{array}\right]=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] $
$\Rightarrow\left[\begin{array}{ll} 48+2 a & 20+2 b \\ 56+2 c & 76+2 d \end{array}\right]=\left[\begin{array}{ll}
0 & 0 \\ 0 & 0 \end{array}\right] $
$\Rightarrow 48+2 a =0$
$ \Rightarrow 2 a =-48$
$ \Rightarrow a =-24= 20+2 b =0 $
$\Rightarrow 2 b =20 $
$\Rightarrow b =-10 = 56+2 c =0$
$ \Rightarrow 2 c =56 $
$\Rightarrow c =-28 = 76+2 d =0 $
$\Rightarrow 2 d =-76 $
$\Rightarrow d =-38$
Thus $C =\left[\begin{array}{ll} -24 & -10 \\ -28 & -38\end{array}\right] .$
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
In an examination the ratio of the number of suo:essful candidates to unsuccessful candidates is 7:5. Had 30 more appeared and 10 more passed, the ratio of successful candidates to unsuccessful candidates would have been 4:3. Find the number of candidates who appeared in the examination originally.
If $\frac{a}{c}=\frac{c}{d}=\frac{c}{f}$ prove that $: \frac{\left(a^3+c^3\right)^2}{\left(b^3+d^3\right)^2}=\frac{e^6}{f^6}$
→If Δ PQR is isosceles with PQ = PR and a circle with centre O and radius r is the incircle of the Δ PQR touching QR at T, prove that the point T bisects QR.
From the top of a lighthouse, the angles of depression of two ships on the opposite sides of it are observed to be $\alpha$, and $\beta$. If the height of the light house is ' $h$ ' $m$ and the line joining the ships passes through the foot of the light house, show that the distance between the ship is $\frac{\mathrm{h}(\tan \alpha+\tan \beta)}{\tan \alpha \tan \beta} \mathrm{m}$
→The following figure shows a circle with PR as its diameter.
If PQ = 7 cm and QR = 3RS = 6 cm, find the perimeter of the cyclic quadrilateral PQRS.
If PQ = 7 cm and QR = 3RS = 6 cm, find the perimeter of the cyclic quadrilateral PQRS.
If $P=\left|\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right|$ and $Q=\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|$ find $P(Q P)$.
→Due to a heavy storm, a part of a banyan tree broke without separating from the main. The top of the tree touched the ground 15 m from the base making an angle of 45° with the ground. Calculate the height of the tree before it was broken.
A manufacturing company ‘P’ sells a Desert cooler to a dealer A for ₹ 8100 including sales tax (under VAT). The dealer A sells it to a dealer B for ₹ 8500 plus sales tax and the dealer B sells it to a consumer at a profit of ₹ 600. If the rate of sales tax (under VAT) is 8%, find
(i) the cost price of the cooler for dealer A.
(ii) the amount of tax received by the Government.
(iii) the amount which the consumer pays for the cooler.
(i) the cost price of the cooler for dealer A.
(ii) the amount of tax received by the Government.
(iii) the amount which the consumer pays for the cooler.
If $a, b, c$ and $d$ are in proportion, prove that:
$
\frac{a^2+a b+b^2}{a^2-a b+b^2}=\frac{c^2+c d+d^2}{c^2-c d+d^2}
$
→$
\frac{a^2+a b+b^2}{a^2-a b+b^2}=\frac{c^2+c d+d^2}{c^2-c d+d^2}
$
Using a ruler and a compass construct a triangle ABC in which AB = 7 cm, ∠CAB = 60o and
AC = 5 cm. Construct the locus of
1) points equidistant from AB and AC
2) points equidistant from BA and BC
Hence construct a circle touching the three sides of the triangle internally.
AC = 5 cm. Construct the locus of
1) points equidistant from AB and AC
2) points equidistant from BA and BC
Hence construct a circle touching the three sides of the triangle internally.