Finding x and y by Equating Matrices
Question:
Find \( x \) and \( y \) if
\[ \begin{bmatrix} x+10 & y^2+2y \\ 0 & -4 \end{bmatrix} = \begin{bmatrix} 3x+4 & 3 \\ 0 & y^2-5y \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ x + 10 = 3x + 4 \quad …(1) \]
\[ y^2 + 2y = 3 \quad …(2) \]
\[ y^2 – 5y = -4 \quad …(3) \]
Step 2: Solve for x
From (1):
\[ x + 10 = 3x + 4 \Rightarrow 2x = 6 \Rightarrow x = 3 \]
Step 3: Solve for y
From (2):
\[ y^2 + 2y – 3 = 0 \Rightarrow (y+3)(y-1)=0 \]
\[ y = -3 \text{ or } y = 1 \]
From (3):
\[ y^2 – 5y + 4 = 0 \Rightarrow (y-4)(y-1)=0 \]
\[ y = 4 \text{ or } y = 1 \]
Step 4: Common Value
The common solution satisfying both equations is:
\[ y = 1 \]
Final Answer
\[ x = 3,\quad y = 1 \]