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:
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:
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In scalar matrices, diagonal elements are constant, and off-diagonal elements are zero.
Updated On: Feb 6, 2025
- \( 0 \)
- \( 5 \)
- \( 10 \)
- \( 25 \)
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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).
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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