Determine which of the following are true and which are false

Question

1. There exists a 3 x 3 matrix whose range is the same as its null space.
2. The rank of a matrix is the number of non-zero rows that it has.
3. For all square matrices A, it's true that det(-A) = -det(A).
4. If A and B are square matrices of the same size, then det(AB) = det(BA)
5. If A and B are square matrices of the same size, then det(A + B) = det(A) + det(B).

Answer 1

1 is true. Consider the matrix [1 -1][1 -1]. The reduced row echelon form of this matrix is [1,-1][0,0]. So, the null space for this matrix are the vectors (x,y) that satisfy the equation x-y=0. Or rather, x=y. This means that the null space is the space spanned by the vector (1,1).

On the other hand, if we multiply our matrix to the vector (x,y), we get the vector (x-y,x-y)=(x-y)(1,1). Because x-y can have any value we see that the range is also the subspace spanned by (1,1).

2 is false. The rank is the number of non zero row in the row echelon form of the matrix.

3 is also false. For any scalar c, det(cA)=cⁿ * det A. So, det(-A)=(-1)ⁿdet A.

4 is true because,det(AB)=det A*det B=det B*det A=det(BA).

5 is false. For example, det I=1, but det (I+I)=det(2I)=2^n which is not equal to 2 if n>1.