Finding Variables by Equating Matrices
Question:
Find \( x, y, a, b \) if
\[ \begin{bmatrix} 2x – 3y & a – b & 3 \\ 1 & x + 4y & 3a + 4b \end{bmatrix} = \begin{bmatrix} 1 & -2 & 3 \\ 1 & 6 & 29 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ 2x – 3y = 1 \quad …(1) \]
\[ a – b = -2 \quad …(2) \]
\[ x + 4y = 6 \quad …(3) \]
\[ 3a + 4b = 29 \quad …(4) \]
Step 2: Solve for x and y
From (3): \( x = 6 – 4y \)
Substitute into (1):
\[ 2(6 – 4y) – 3y = 1 \]
\[ 12 – 8y – 3y = 1 \Rightarrow 12 – 11y = 1 \Rightarrow 11y = 11 \Rightarrow y = 1 \]
\[ x = 6 – 4(1) = 2 \]
Step 3: Solve for a and b
From (2): \( a = b – 2 \)
Substitute into (4):
\[ 3(b – 2) + 4b = 29 \]
\[ 3b – 6 + 4b = 29 \Rightarrow 7b = 35 \Rightarrow b = 5 \]
\[ a = 5 – 2 = 3 \]
Final Answer
\[ x = 2,\quad y = 1,\quad a = 3,\quad b = 5 \]