If A= 2&5 2&1 and B= 4&-3 2&5 , verify that |AB| = |A| |B|. - Vidyadip

Question

If $\text{A}=\begin{vmatrix}2&5\\2&1\end{vmatrix}$ and $\text{B}=\begin{vmatrix}4&-3\\2&5\end{vmatrix},$ verify that |AB| = |A| |B|.

✓

Answer

Let $\text{A}=\begin{vmatrix}2&5\\2&1\end{vmatrix}$
$\Rightarrow|\text{A}|=2-10=-8$
$\text{B}=\begin{vmatrix}4&-3\\2&5\end{vmatrix}$
$\Rightarrow|\text{B}|=20+6=26$
Now $\text{AB}=\begin{vmatrix}2&5\\2&1\end{vmatrix}\begin{vmatrix}4&-3\\2&5\end{vmatrix}$
$=\begin{vmatrix}2\times4+5\times2&2\times(-3)+5\times5\\2\times4+1\times2&2\times(-3)+1\times5\end{vmatrix}$
$=\begin{vmatrix}8+10&-6+25\\8+2&-6+5\end{vmatrix}$
$=\begin{vmatrix}18&19\\10&-1\end{vmatrix}$
$\Rightarrow|\text{AB}|=18\times(-1)-(10)(19)$
$=-18-190=-208$
Now $|\text{AB}|=|\text{A}|\times|\text{B}|$
$-208=(-8)\times(26)$
$-208=-208$
Hence verified.

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

Show that the points A (1, -2, -8), B (5, 0, -2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the coordinates of the point.

Verify Rolle's theorem for the following function on the indicated intervals$f(x) = x(x^- 4)^2$ on the interval $[0, 4]$

Find the shortest distance between the lines $I_1$ and $l_2$ $\vec{r}=(-\hat{\imath}-\hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-6 \hat{\jmath}+\hat{k})$ and $\vec{r}=(3 \hat{\imath}+5 \hat{\jmath}+7 \hat{k})+\mu(\hat{\imath}-2 \hat{\jmath}+\hat{k})$where $\lambda$ and $\mu$ are parameters.

A wire of length 20m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum ?

An item is manufactured by three machines $A , B$ and $C$ . Out of the total number of items manufactured during a specified period, $50 \%$ are manufactured on $A , 30 \%$ on B and $20 \%$ on C. $2 \%$ of the items produced on A and $2 \%$ of items produced on B are defective, and $3 \%$ of these produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Consider $\text{f : R} - \left\{-\frac{4}{3}\right\} \rightarrow \text{R} - \left\{\frac{4}{3}\right\} \text{given by f(x)} = \frac{\text{4x + 3}}{\text{3x + 4}}.$ Show that f is bijective. Find the inverse of f and hence find $f ^{–1}(0)$ and x such that $f ^{–1}(x) = 2$.

A salesman has the following record of sales during three months for three items A, B and C which have different rates of commission.

Month	Sale of units			Total commission drawn (in Rs.)
	A	B	C
Jan	90	100	20	800
Feb	130	50	40	900
March	60	100	30	850

Find out the rates of commission on items A, B and C by using determinant method.

Using elementary row operations (transformations), find the inverse of the following matrix:
$\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 0 \end{bmatrix}$

Find the angle between the vectors whose direction cosines are proportional to 2, 3, -6 and 3, -4, 5.