Finding x, y, z, w by Equating Matrices
Question:
Find \( x, y, z, w \) if
\[ \begin{bmatrix} xy & 4 \\ z + 6 & x + y \end{bmatrix} = \begin{bmatrix} 8 & w \\ 0 & 6 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ xy = 8 \quad …(1) \]
\[ 4 = w \quad …(2) \]
\[ z + 6 = 0 \quad …(3) \]
\[ x + y = 6 \quad …(4) \]
Step 2: Solve for x and y
From (4): \( x + y = 6 \)
From (1): \( xy = 8 \)
So, \( x \) and \( y \) are roots of:
\[ t^2 – 6t + 8 = 0 \]
\[ (t – 2)(t – 4) = 0 \Rightarrow t = 2 \text{ or } 4 \]
Thus, \( x = 2, y = 4 \) or \( x = 4, y = 2 \)
Step 3: Solve for z and w
From (3):
\[ z = -6 \]
From (2):
\[ w = 4 \]
Final Answer
\[ (x, y) = (2,4) \text{ or } (4,2),\quad z = -6,\quad w = 4 \]