Let M=left{A=begin{bmatrix} a & b c & d end{ | vedclass

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

Question

(A) We are given $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ where $a, b, c, d \in \{\pm 3, \pm 2, \pm 1, 0\}$.We need to find the number of ma

Vedclass · Accepted Answer

$16$

Vedclass · Answer

(A) We are given $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$ where $a, b, c, d \in \{\pm 3, \pm 2, \pm 1, 0\}$.We need to find the number of matrices such that $\det(A) = ad - bc = 15$.Since the maximum value of $ad$ is $3 	imes 3 = 9$ and the minimum value of $bc$ is $-3 	imes 3 = -9$,the maximum value of $ad - bc$ is $9 - (-9) = 18$.Possible values for $ad$ and $bc$ such that $ad - bc = 15$:Case $I$: $ad = 9$ and $bc = -6$.For $ad = 9$,the pairs $(a, d)$ can be $(3, 3)$ or $(-3, -3)$ ($2$ pairs).For $bc = -6$,the pairs $(b, c)$ can be $(3, -2), (-3, 2), (-2, 3), (2, -3)$ ($4$ pairs).Total matrices for Case $I = 2 	imes 4 = 8$.Case $II$: $ad = 6$ and $bc = -9$.For $ad = 6$,the pairs $(a, d)$ can be $(3, 2), (2, 3), (-3, -2), (-2, -3)$ ($4$ pairs).For $bc = -9$,the pairs $(b, c)$ can be $(3, -3), (-3, 3)$ ($2$ pairs).Total matrices for Case $II = 4 	imes 2 = 8$.Total number of such matrices $= 8 + 8 = 16$.

Similar Questions

If the system of equations $ax + y + z = 0$,$x + by + z = 0$ and $x + y + cz = 0$,where $a, b, c \neq 1$,has a non-trivial solution,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ is

If $\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1+y & 1+2 y & 1 \\ 1+z & 1+z & 1+3 z\end{array}\right| = 10 k x y z \left(3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$,then $k = \text{ . . . . . . }$ (where $x, y, z \neq 0$ and $3+\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \neq 0$).

Let $A = \begin{bmatrix} x+2 & 3x \\ 3 & x+2 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \\ 5 & x+2 \end{bmatrix}$. Then all solutions of the equation $\det(AB) = 0$ are:

If $a, b, c$ are positive and unequal,show that the value of the determinant $\Delta = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ is negative.

Find the area of the triangle with vertices $(a \cos \theta, b \sin \theta)$,$(-a \sin \theta, b \cos \theta)$,and $(-a \cos \theta, -b \sin \theta)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module