Question

Math 2890 QZ-6 SP 2018 1) Find the rank of the following matrix. Also find a basis for the row and column spaces. 1 0 3 3 10 0 -1 2 Find a basis of Null(A) where A is the given matrix. Find the rank of A and dimension of Nul(A). Let B be an invertible 4X4 matrix (a matrix with 4 rows and 4 columns). Is the matrix AATB also invertible? Explain.

Answer 1

the given matrix is   $\begin{bmatrix} 1 &0 &3 &0 \\3 &1 &10 &0 \\ -1 &1 &-2 & 1\\ 1 &-1 &2 &-2 \end{bmatrix}$

reduced the matrix to row echlon form

R₂$\rightarrow$R₂-3R₁ , R₃$\rightarrow$R₃+R₁ ,R₄$\rightarrow$R₄-R₁ we have

$\begin{bmatrix} 1 &0 &3 &0 \\0 &1 &1 &0 \\0 &1 &1 & 1\\ 0 &-1 &-1 &-2 \end{bmatrix}$

apply R₃$\rightarrow$R₃-R₂ ,R₄$\rightarrow$R₄+R₂

$\begin{bmatrix} 1 &0 & 3 &0 \\0 &1 &1 &0 \\0 &0 &0 &1 \\0 &0 &0 &-2 \end{bmatrix}$

R₄$\rightarrow$R₄+2R₃

$\begin{bmatrix} 1 &0 &3 &0 \\0 &1 &1 &0 \\0 &0 &0 &1 \\0 &0 &0 &0 \end{bmatrix}$

which is the row echlon form having one complete zerro row

hence its nullity=1 and rank=4-1=3

## the matrix AA^TB is not invertible if B is invertible

{ a matrix is invertible if its determinant is not zero}

as rank of A is 3<4=order of the matrix

hence determinant of A=0

so det(AA^TB)=det(A)det(A^TB)

=0.det(A^TB)=0

hence it is not invertible