Answer to Question #288347 in Linear Algebra for Sabelo Xulu

Question

Q10.

For a given matrix A = $\begin{bmatrix} 1 & 0 & 2 & -3 \\ 2 & 0 & 4 & - 6 \\ -3 & 0 & - 6& 9 \end{bmatrix}$. Which of the following is true

(1) rank (A) = 3, nullity (A) = 1

(2) rank (A) = 2, nullity (A) = 2

(3) rank (A) = 1, nullity (A) = 3

(4) None of the given answers is true.

Q11.

For a, b $\in$ R, the transformation T : R^2 $\to$ R^3 defined byT (x, y) = (2x - y, 3x + y + 3A, 5x - 2y + bxy) is linear if

(1) a = b = 1

(2) a = 0; b = 1

(3) a = 1; b = 0

(4) None of the given answers is true.

Q12.

Suppose T : R^3 $\to$ R^2

is a linear defined by

T (x, y) = (4x + 3y + Z, x - 2y). Then which of the following is basis of range T is

(1) (4, 1, 3), (1, 2, 0)

(2) (1, 0), (0, 1), (1, 1)

(3) (4, 1), (3, 2), (1, 0)

(4) None of the given answers is true.

Answer 1

Q 10

Reduced row Echolon form of A

$rref\>A=\begin{pmatrix} 1&0&2&-3 \\ 0&0&0&0 \\ 0&0&0&0 \end{pmatrix}$

Rank (A)= 1, Nullity (A) =3

Q 11

If T is linear it should satisfy

T(cx,cy)=cT(x,y)

For any scalar c, and any element x,y in the domain of T.

T(cx,cy)= (2cx-cy, 3cx+cy+3a, 5cx-2cy+bcxcy)

$\ne$ c(2x-y, 3x+y+3a, 5x-2y+bxy)

None of the given answer is true.

Q 12

$\begin{pmatrix} 4x+3y+z \\ x-2y \end{pmatrix}=x\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ $+y\begin{pmatrix} 3 \\ -2 \end{pmatrix}+z\begin{pmatrix} 1 \\ 0 \end{pmatrix}$

Transforming matrix

$A=\begin{pmatrix} 4&3&1 \\ 1&-2 & 0\\ \end{pmatrix}$

rref A $=\begin{pmatrix} 1&0&\frac{2}{11} \\ 0&1&\frac{1}{11} \end{pmatrix}$

Basis for range T is (1,0),(0,1)

None of the given answers is true.