Answer:
To find the rank and nullity of A, we need to find the row echelon form of A. After performing Gaussian elimination, we get:
⎡⎣
1 -3/4 0 -1/4
0 1 -10 3
0 0 0 0
0 0 0 0
0 0 0 0
⎤⎦
The non-zero rows of the row echelon form of A correspond to the rows of A that contain the pivots. Therefore, the rank of A is 2.
To find the nullity of A, we need to find all solutions to the equation Ax = 0, where 0 is a vector of zeros and x is a vector of unknowns. We can solve this equation using Gaussian elimination and back-substitution on the row echelon form of A:
x1 = 3/4x4
x2 = 10x4 - 3x3
x3 = free
x4 = free
Thus, the general solution to Ax = 0 is:
x = t1 * ⎡⎣
-3/4
3
0
1
⎤⎦
+ t2 * ⎡⎣
-3/4
0
1
0
⎤⎦
where t1 and t2 are any constants.
Therefore, the nullity of A is 2.
By the Dimension Theorem, we have:
rank(A) + nullity(A) = 2 + 2 = 4 = number of columns of A
This verifies that the values we obtained for the rank and nullity of A satisfy Formula (4) in the Dimension Theorem.