If k is a scalar and I is the unit matrix of order 3, then a d j(k I) =

• A: −k₃I
• B: −k₂I
• C: k₂I
• D: k₃I

To solve the problem involving the adjugate of a scalar multiplied by the unit matrix, we start by recalling some properties of matrices.

Understanding the Adjugate

The adjugate of a matrix, denoted as adj(A), is related to the determinant and the inverse of the matrix. For a scalar \( k \) and the unit matrix \( I \) of order 3, we can express this as:

Calculating the Adjugate

The unit matrix \( I \) is a diagonal matrix with ones on the diagonal and zeros elsewhere. When we multiply \( I \) by a scalar \( k \), we get:

• kI =

          | k 0 0 |
          | 0 k 0 |
          | 0 0 k |
          

The adjugate of a scalar multiple of a matrix can be calculated using the formula:

adj(kI) = k^(n-1) * I, where n is the order of the matrix.

Applying the Formula

Since our matrix is of order 3 (n = 3), we have:

adj(kI) = k^(3-1) * I = k² * I.

Final Answer

Thus, the correct answer is:

• C: k²I