Question

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:

• A. \( 0 \)
• B. \( 5 \)
• C. \( 10 \)
• D. \( 25 \)

Hint:

In scalar matrices, diagonal elements are constant, and off-diagonal elements are zero.

Answer 1

The Correct Option is D

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).