Let A= 3 & 2 7 & 5 and B= 6 & 7 8 & 9 . Find (AB)^ -1 . - Vidyadip

Question

Let $\text{A}=\begin{bmatrix}3 & 2 \\7 & 5 \end{bmatrix}\text{and B}=\begin{bmatrix}6 & 7 \\8 & 9 \end{bmatrix}$. Find $(AB)^{-1}$.

✓

Answer

$\text{A}=\begin{bmatrix}3 & 2\\7 & 5 \end{bmatrix}\therefore\ |\text{A}|=1\neq0\text{ and adj A}=\begin{bmatrix}5 & -2 \\-7 & 3 \end{bmatrix}$
$\therefore\ \text{A}^{-1}\frac{\text{adj A}}{|\text{A}|}=\frac{1}{1}\begin{bmatrix}5 & -2 \\-7 & 3 \end{bmatrix}$
$\text{B}=\begin{bmatrix}6 & 7 \\7 & 9 \end{bmatrix}\therefore\ |\text{B}|=-2\neq0\text{ and adj B}=\begin{bmatrix}9 & -7 \\-8 & 6 \end{bmatrix}$
$\therefore\ \text{B}^{-1}=\frac{\text{adj B}}{|\text{B}|}=\frac{1}{(-2)}=\begin{bmatrix}9 & -7 \\-8 & 6 \end{bmatrix}$
Now, $(\text{AB})^{-1}=\text{B}^{-1}\text{A}^{-1}$
$(\text{AB})^{-1}=\frac{1}{(-2)}\begin{bmatrix}9 & -7 \\-8 & 6 \end{bmatrix}\begin{bmatrix}5 & -2 \\-7 & 3 \end{bmatrix}$
$(\text{AB})^{-1}=-\frac{1}{2}\begin{bmatrix}94 & -39 \\-82 & 34 \end{bmatrix}$
$\text{(AB)}^{-1}=\begin{bmatrix}-47 & \frac{39}{2} \\41 & -17 \end{bmatrix}$

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

Find the points on the curve $y = 3x^2 - 9x + 8$ at which the tangents are equally inclined with the axes.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $\frac{4\text{r}}{3}.$ Also show that the maximum volume of the cone is$\frac{8}{27}$of the volume of the sphere.

Evaluate the following integrals:
$\int\frac{\text{x}^2+1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$

For the matrix $\text{A}=\begin{bmatrix}1&1&1\\1&2&-3\\2&-1&3\end{bmatrix}$Show that $A^3 - 6A^2 + 5A + 11I = 0$. Hence, find $A^{-1}.$

Let $A =\{1,2,3\}$ and $R =\left( a , b ): a , b \in A\right.$ and $\left|a^2-b^2\right| \leq 5$. Write $R$ as set of ordered pairs. Mention whether $R$ is $i.$ reflexive
$ii$. symmetric
$iii$. transitive
Give reason in each case.

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10,000. If the profit is Rs. 60 per unit for the product A and Rs. 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.

A ladder $5$ m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2$ cm/s. How fast is its height on the wall decreasing when the foot of the ladder is $4$ m away from the wall?

Differentiate $\cos^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big),$ if $0<\text{x}<1$

Solve the following differential equation:
$\text{y dx}+\Big\{\text{x}\log\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}-2\text{x dy}=0$

If $\text{A}=\begin{bmatrix}1&2\\2&1\end{bmatrix}, f(x) = x^2 - 2x - 3$, show that $f(A) = 0$