Finding x, y, z, w by Equating Matrices
Question:
Find \( x, y, z, w \) if
\[ \begin{bmatrix} x & 3x – y \\ 2x + z & 3y – w \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ x = 3 \quad …(1) \]
\[ 3x – y = 2 \quad …(2) \]
\[ 2x + z = 4 \quad …(3) \]
\[ 3y – w = 7 \quad …(4) \]
Step 2: Solve for x and y
From (1): \( x = 3 \)
Substitute into (2):
\[ 3(3) – y = 2 \Rightarrow 9 – y = 2 \Rightarrow y = 7 \]
Step 3: Solve for z
From (3):
\[ 2(3) + z = 4 \Rightarrow 6 + z = 4 \Rightarrow z = -2 \]
Step 4: Solve for w
From (4):
\[ 3(7) – w = 7 \Rightarrow 21 – w = 7 \Rightarrow w = 14 \]
Final Answer
\[ x = 3,\quad y = 7,\quad z = -2,\quad w = 14 \]