A determinant is chosen at random from the se | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) $2 	imes 2$ determinant has $4$ positions,each can be filled by $0$ or $1$. Thus,the total number of possible determinants is $2^4 = 16$.Let the

Vedclass · Accepted Answer

$\frac{3}{8}$

Vedclass · Answer

(B) $2 	imes 2$ determinant has $4$ positions,each can be filled by $0$ or $1$. Thus,the total number of possible determinants is $2^4 = 16$.Let the determinant be $\begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$.The determinant is non-zero if $ad - bc 
eq 0$,which means $ad 
eq bc$.Since $a, b, c, d \in \{0, 1\}$,the possible values for $ad$ and $bc$ are $0$ or $1$.The condition $ad 
eq bc$ holds in the following cases:$1$. $ad = 1$ and $bc = 0$: This implies $a=1, d=1$ and $(b=0$ or $c=0)$. The pairs $(b, c)$ can be $(0, 0), (0, 1), (1, 0)$. There are $3$ such matrices.$2$. $ad = 0$ and $bc = 1$: This implies $b=1, c=1$ and $(a=0$ or $d=0)$. The pairs $(a, d)$ can be $(0, 0), (0, 1), (1, 0)$. There are $3$ such matrices.Total number of matrices with non-zero determinant $= 3 + 3 = 6$.Therefore,the required probability $= \frac{6}{16} = \frac{3}{8}$.

$A$ determinant is chosen at random from the set of all determinants of order $2 \times 2$ with elements $0$ or $1$ only. The probability that the determinant chosen is non-zero is .........

Similar Questions

If $A, B, C$ are the angles of a triangle,then the system of equations $-x + y \cos C + z \cos B = 0$,$x \cos C - y + z \cos A = 0$,$x \cos B + y \cos A - z = 0$ have

The characteristic equation of the matrix $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 2 & 5 \\ 3 & -1 & 2 \end{bmatrix}$ is:

If $q_1, q_2, q_3$ are roots of the equation $x^3 + 64 = 0$,then the value of $\left| \begin{array}{ccc} q_1 & q_2 & q_3 \\ q_2 & q_3 & q_1 \\ q_3 & q_1 & q_2 \end{array} \right|$ is

If $p + q + r = 0$ and $a + b + c = 0$,then the value of the determinant $\left| \begin{array}{ccc} pa & qb & rc \\ qc & ra & pb \\ rb & pc & qa \end{array} \right|$ is

If $\begin{bmatrix} 1 & 2 & -1 \\ 1 & x-2 & 1 \\ x & 1 & 1 \end{bmatrix}$ is a singular matrix,then the value of $x$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module