Linear Algebra Answers

Show that T(x1, x2, x3, x4) = 3x1 −7x2 + 5x4 is a linear transformation by finding the

matrix for the transformation. Then find a basis for the null space of the transforma￾tion.

1.Solve for X from the matrix equation below. Here l is the identity matrix and det(B) ≠ 0 and det(A) ≠ 0.

B(X - l)A + B = A

Choose the correct option:

1. No such matrix X.

2. X=A–¹ B - A.

3. X=A–¹ - B +A.

4. X=A–¹ + B -A.

5. X= -A–¹ + B–¹ + l.

6. X= -(A–¹ + B–¹).

2. Consider the following linear system:

2x - 3y = -1

2x - 3y = 1

1. x=0 and y= 0 satisfy the system.

2. The system has exactly one solution.

3. The system is inconsistent.

4. The system has infinitely many solution.

1.Which of the following is a linear equation in x,yand z?

1. -x–¹ + e–√²y=3z l, where e= 2.71828

2. 2πln(e–½) - 2y + z= ln(3) - x.

3. √y² + 4y - 2z = 7x.

4.y + 4y - 2z = 7x–².

2.Solve for X from the matrix equation below. Here is the identity matrix and det(A) ≠ 0.

A²X + l = AB

Choose the correct option:

1. X is the identity matrix.

2. X= A–¹B - l

3. X= A–¹B + l

4. X= A–¹B - A

5. X= AB + B

{F} Work Out

1. Solve by crammer's rule, the following equations.

3X + 3Y - Z = 11

2X - Y + 2Z = 9

4X + 3Y - 27 = 25

2. solve the above equations using graphical method. 

Determine which of the following is not the solution set of the linear equations below.

3x-y+z=2

2x-z=2

Let A=[a b c         ←Matrix

d e f

g h i]

where a, b, c, d,e, f, g, h, i are some real numbers, if det(A)=5 answer the following questions:

1. Is A invertible? (Justify your answer). Find rank(A)

1. Let b= [a+d b+e c+f         ←Matrix

d      e       f

2g     2h    2i]

And, C= [a b c                   ←Matrix

     -2d -2e -2f

     3g 3h 3I ]

Compute det(B) and det(C).

(C) Compute det(A^-1) and det(adj(A)).

<e> Show that if P and Q are vector Subspaces of a vector space V then P ∩ Q is also a vector subspace of V.