Finding Variables by Equating Matrices
Question:
Find \( x, y, a, b \) if
\[ \begin{bmatrix} 3x + 4y & 2 & x – 2y \\ a + b & 2a – b & -1 \end{bmatrix} = \begin{bmatrix} 2 & 2 & 4 \\ 5 & -5 & -1 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ 3x + 4y = 2 \quad …(1) \]
\[ x – 2y = 4 \quad …(2) \]
\[ a + b = 5 \quad …(3) \]
\[ 2a – b = -5 \quad …(4) \]
Step 2: Solve for x and y
From (2): \( x = 4 + 2y \)
Substitute into (1):
\[ 3(4 + 2y) + 4y = 2 \]
\[ 12 + 6y + 4y = 2 \Rightarrow 12 + 10y = 2 \Rightarrow 10y = -10 \Rightarrow y = -1 \]
\[ x = 4 + 2(-1) = 2 \]
Step 3: Solve for a and b
From (3): \( b = 5 – a \)
Substitute into (4):
\[ 2a – (5 – a) = -5 \]
\[ 2a – 5 + a = -5 \Rightarrow 3a = 0 \Rightarrow a = 0 \]
\[ b = 5 – 0 = 5 \]
Final Answer
\[ x = 2,\quad y = -1,\quad a = 0,\quad b = 5 \]