Question

The number of all scalar matrices of order 3, with each entry \( -1, 0, 1 \), is:

• A. \( 1 \)
• B. \( 3 \)
• C. \( 2 \)
• D. \( 3^9 \)

Hint:

A scalar matrix is always a diagonal matrix where all diagonal elements are equal. The off-diagonal elements are always zero, simplifying calculations and classification.

Answer 1

The Correct Option is B

Step 1: Understanding scalar matrices. A scalar matrix is a special type of diagonal matrix in which all diagonal elements are equal, and all off-diagonal elements are zero. For a \( 3 \times 3 \) scalar matrix, the structure is: \[ \begin{bmatrix} a & 0 & 0
0 & a & 0
0 & 0 & a \end{bmatrix}, \] where \( a \) is the scalar value for all diagonal entries. Step 2: Determine the possible values for \( a \). According to the problem, the entries in the matrix can take the values \( -1, 0, \) or \( 1 \). Since \( a \) must be the same for all diagonal entries, there are exactly three possibilities for \( a \): \[ a = -1, \quad a = 0, \quad a = 1. \] Step 3: Identify the corresponding scalar matrices. The three scalar matrices corresponding to these values of \( a \) are: \[ \begin{bmatrix} -1 & 0 & 0
0 & -1 & 0
0 & 0 & -1 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0 & 0
0 & 0 & 0
0 & 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 1 & 0 & 0
0 & 1 & 0
0 & 0 & 1 \end{bmatrix}. \] Thus, there are exactly 3 scalar matrices of order 3 with the specified values for the entries.