Question

Let 

and \( I_3 \) be the 3 × 3 identity matrix. Then the nullity of \( 5A(I_3 + A + A^2) \) equals _________

Hint:

The nullity of a matrix is the number of free variables in its solution to the homogeneous equation \( A \mathbf{x} = 0 \).

Answer 1

Correct Answer: 2

Step 1: Compute \( I_3 + A + A^2 \). 
We begin by calculating the powers of matrix \( A \): 

Thus, \[ I_3 + A + A^2 = I_3 + A + I_3 = 2I_3 + A. \] Step 2: Multiply by 5A: \[ 5A(I_3 + A + A^2) = 5A(2I_3 + A) = 10A + 5A^2. \] Since \( A^2 = I_3 \), \[ 5A(I_3 + A + A^2) = 10A + 5I_3. \] Step 3: Find the nullity. The nullity of a matrix is the dimension of the null space. Since the matrix \( 10A + 5I_3 \) is a linear combination of \( A \) and \( I_3 \), it has a nullity of 1. Thus, the nullity is \( 1 \).