If
\[
\begin{bmatrix}
a & c & 0 \\
b & d & 0 \\
0 & 0 & 5
\end{bmatrix}
\]
is a scalar matrix, then the value of \( a + 2b + 3c + 4d \) is:
Show Hint
- \( 0 \)
- \( 5 \)
- \( 10 \)
- \( 25 \)
The Correct Option is D
Solution and Explanation
Step 1: Definition of a scalar matrix
A scalar matrix is a diagonal matrix where all diagonal elements are equal.
This means \( a = d = 5 \), and all off-diagonal elements (\( b \), \( c \)) are zero.
Step 2: Substitute the values
Using the scalar matrix properties:
\[
a + 2b + 3c + 4d = 5 + 2(0) + 3(0) + 4(5) = 5 + 0 + 0 + 20 = 25.
\]
Step 3: Verify the options
The correct value is \( 25 \), which corresponds to option (D).
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