Finding x, y, z, w by Equating Matrices
Question:
Find \( x, y, z, w \) if
\[ \begin{bmatrix} x – y & z \\ 2x – y & w \end{bmatrix} = \begin{bmatrix} -1 & 4 \\ 0 & 5 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ x – y = -1 \quad …(1) \]
\[ z = 4 \quad …(2) \]
\[ 2x – y = 0 \quad …(3) \]
\[ w = 5 \quad …(4) \]
Step 2: Solve for x and y
Subtract (1) from (3):
\[ (2x – y) – (x – y) = 0 – (-1) \]
\[ x = 1 \]
Substitute into (1):
\[ 1 – y = -1 \Rightarrow y = 2 \]
Step 3: Write Remaining Values
\[ z = 4,\quad w = 5 \]
Final Answer
\[ x = 1,\quad y = 2,\quad z = 4,\quad w = 5 \]