Finding x and y by Equating Matrices
Question:
Find \( x \) and \( y \) if
\[ A = \begin{bmatrix} 2x+1 & 2y \\ 0 & y^2 – 5y \end{bmatrix}, \quad B = \begin{bmatrix} x+3 & y^2 + 2 \\ 0 & -6 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ 2x + 1 = x + 3 \quad …(1) \]
\[ 2y = y^2 + 2 \quad …(2) \]
\[ y^2 – 5y = -6 \quad …(3) \]
Step 2: Solve for x
From (1):
\[ 2x – x = 3 – 1 \Rightarrow x = 2 \]
Step 3: Solve for y
From (3):
\[ y^2 – 5y + 6 = 0 \Rightarrow (y – 2)(y – 3) = 0 \]
\[ y = 2 \quad \text{or} \quad y = 3 \]
Step 4: Check Consistency
Substitute into (2):
For \( y = 2 \): \( 2y = 4,\; y^2 + 2 = 6 \) ❌ Not equal
For \( y = 3 \): \( 2y = 6,\; y^2 + 2 = 11 \) ❌ Not equal
Conclusion
No value of \( y \) satisfies all equations simultaneously.
Final Answer
No solution exists (matrices are not equal for any value of \( y \)).