Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
$A = \begin{bmatrix} \frac{1}{2}[x] & |\sin y| \\ \cos z & 1 \end{bmatrix}$, $B = \begin{bmatrix} [x] & [y] \\ [z] & 1 \end{bmatrix}$ if $x \in [-2, 2]$, $y, z \in (-\pi, \pi)$ if number of triplets $(x, y, z)$ such that $A = B$ is $k$, then value of $k/7$ is _____.
Step-by-Step Solution
Key Concept: For matrices A and B to be equal, corresponding elements must match: $\frac{1}{2}[x] = [x]$, $|\sin y| = [y]$, $\cos z = [z]$, and $1 = 1$. This requires finding values where floor functions equal fractional/trigonometric expressions using properties of floor function $[x]$ and fractional part $\{x\} = x - [x]$.
For $\frac{1}{2}|x| = \{x\}$ with $x \in [-2,2]$, we find $x = 0, -\frac{1}{3}, \frac{3}{2}$. For $\sin y = \{y\}$ with $y \in (-\pi, \pi)$, $y$ can take 7 values. For $\cos z = \{z\}$ with $z \in (-\pi, \pi)$, $z$ can take 2 values. Therefore $(x,y,z)$ can take $3 \times 7 \times 2 = 42$ triplets using graphical analysis.
Correct Answer: 6