Matrix Element Problem Solution
Question:
If \( A = [a_{ij}] = \begin{bmatrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{bmatrix} \) and \( B = [b_{ij}] = \begin{bmatrix} 2 & -1 \\ -3 & 4 \\ 1 & 2 \end{bmatrix} \), find:
- (i) \( a_{22} + b_{21} \)
- (ii) \( a_{11}b_{11} + a_{22}b_{22} \)
Concept
In matrix notation, \( a_{ij} \) represents the element in the i-th row and j-th column. :contentReference[oaicite:0]{index=0}
Step 1: Identify Required Elements
From matrix \( A \):
\[ A = \begin{bmatrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{bmatrix} \]
- \( a_{22} = 4 \)
- \( a_{11} = 2 \)
From matrix \( B \):
\[ B = \begin{bmatrix} 2 & -1 \\ -3 & 4 \\ 1 & 2 \end{bmatrix} \]
- \( b_{21} = -3 \)
- \( b_{11} = 2 \)
- \( b_{22} = 4 \)
Step 2: Compute Values
(i)
\[ a_{22} + b_{21} = 4 + (-3) = 1 \]
(ii)
\[ a_{11}b_{11} + a_{22}b_{22} = (2)(2) + (4)(4) = 4 + 16 = 20 \]
Final Answer
(i) \( 1 \)
(ii) \( 20 \)