If B = [bij] is a scalar matrix of n  n order such that bii = a for all i = 1, 2, …, n, then the value of determinant Bis equal to

1. Break Down the Problem

We need to find the determinant of a scalar matrix $B$ of size $n \times n$, where all diagonal elements $b_{ii} = a$ for $i = 1, 2, \ldots, n$, and all off-diagonal elements $b_{ij} = 0$ for $i \neq j$.

2. Relevant Concepts

A scalar matrix is of the form: $B = aI_n$ where $I_n$ is the identity matrix of size $n \times n$. The determinant of a scalar matrix can be calculated using the property of determinants.

3. Analysis and Detail

The determinant of a scalar matrix can be calculated using the property: $\text{det}(aI) = a^n$ where:

• $a$ is the scalar value on the diagonal,
• $n$ is the order of the matrix.

4. Verify and Summarize

Since $B$ is a scalar matrix defined as $B = aI_n$, we can directly apply the determinant property. Hence: $\text{det}(B) = a^n$

Final Answer

The value of the determinant of matrix $B$ is: $\text{det}(B) = a^n$