Finding a and b by Equating Matrices
Question:
Find \( a \) and \( b \) if
\[ A = \begin{bmatrix} a+4 & 3b \\ 8 & -6 \end{bmatrix}, \quad B = \begin{bmatrix} 2a+2 & b^2+2 \\ 8 & b^2-5b \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ a + 4 = 2a + 2 \quad …(1) \]
\[ 3b = b^2 + 2 \quad …(2) \]
\[ -6 = b^2 – 5b \quad …(3) \]
Step 2: Solve for a
From (1):
\[ a + 4 = 2a + 2 \Rightarrow a = 2 \]
Step 3: Solve for b
From (2):
\[ b^2 – 3b + 2 = 0 \Rightarrow (b-1)(b-2)=0 \]
\[ b = 1 \text{ or } b = 2 \]
From (3):
\[ b^2 – 5b + 6 = 0 \Rightarrow (b-2)(b-3)=0 \]
\[ b = 2 \text{ or } b = 3 \]
Step 4: Common Value
The common value satisfying both equations is:
\[ b = 2 \]
Final Answer
\[ a = 2,\quad b = 2 \]