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MAL101:: Tutorial 1 :: Linear Algebra

(1) Suppose we have a system of three linear equations in real coefficients and in two unknowns:

a1x + b1y = c1

a2x + b2y = c2

a3x + b3y = c3

Interpret geometrically the following statements:

(a) the system has no solutions.(b) the system has a unique solution.(c) the system has an infinitely many solutions.

Further, provide values to ai, bi, ci R (i = 1, 2, 3) so that the above statements hold.

(2) Suppose we have a system of three linear equations in real coefficients and in three unknowns:

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Interpret geometrically the following statements:

(a) the system has no solutions.(b) the system has a unique solution.(c) the system has an infinitely many solutions.

Further, provide values to ai, bi, ci, di (i = 1, 2, 3) so that the above statements hold.

(3) Suppose we have a system of three linear equations in two unknowns:

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

Can this system have a unique solution? Interpret geometrically the following statements:(a) the system has no solutions.

(b) the system has an infinitely many solutions.Further, provide values to ai, bi, ci, di (i = 1, 2) so that the above statements hold.

(4) Suppose we have a system of linear equations in complex coefficients. Can we interpret thestatements in question 1, 2 and 3 exactly in the same way? Solve the following system ofequations (for finding x, y in C ):

(1 i)x + (1 + i)y = 2 + 3i

(1 + i)x + (1 i)y = 3 i

(5) Suppose the lines L1 and L2 are defined byx1

1 =y2

2 =z3

3 andx2

1 =y3

2 =z1

3 . Show thatL1 L2 is empty. Write a system of four linear equations in three unknowns in standardform which has no solutions using the fact that L1 L2 is empty.

(6) Suppose A, B are square matrices of same size. Prove the following statements(a) tr(A + B) = tr(A) + tr(B) and tr(AB) = tr(BA).(b) det(AB) = det(A) det(B) (assume A and B are 2 2 matrices) and in particular,

det(AB) = det(BA).(c) det(A + B) = det(A) + det(B) is false.

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(7) Which of the following matrices are row reduced echelon matrix. Give a reason when thematrix is not row reduced echelon.

1 0 50 2 3

0 0 0

,

1 0 50 1 3

0 0 1

,

0 1 21 0 4

0 0 0

,

0 1 00 0 0

0 0 1

,

0 1 00 0 1

0 0 0

.

(8) Find row reduce echelon matrix row equivalent to the matrices in the previous question.(9) Show that every elementary matrix is invertible and the inverse is an elementary matrix.

(10) Compute the rank of the following matrices. Determine which are invertible.

1 1 22 3 83 1 2

, 1 2 41 1 52 7 3

, 1 3 41 5 13 13 6

.(11) Find inverse of the invertible matrices in the previous question by reducing the matrix to row

reduced echelon form (identity matrix).(12) Write the following matrices as the product of elementary matrices (whenever possible):

1 32 4

,

1 2 30 1 4

0 0 1

, 1 1 22 3 83 1 2

.

(13) Solve the following system of homogeneous linear equations by reducing the coefficient matrix

into row reduced echelon form:a) 2x1 + 4x2 5x3 + 3x4 = 0, 3x1 + 6x2 7x3 + 4x4 = 0, 5x1 + 10x2 11x3 + 6x4 = 0b) x 2y 3z = 0, 2x + y + 3z = 0, 3x 4y 2z = 0,c) x1+2x2+3x32x4+4x5 = 0, 2x1+4x2+8x3+x4+9x5 = 0, 3x1+6x2+13x3+4x4+14x5 = 0.

(14) Solve the following systems of equations by reducing the augmented matrix to the row re-duced echelon form:a) x1 x2 + 2x3 = 1, 2x1 + 2x3 = 1, x1 3x2 + 4x3 = 2,b) x1 + 7x2 + x3 = 4, x1 2x2 + x3 = 0, 4x1 + 5x2 + 9x3 = 9,c) x2 + 5x3 = 4, x1 + 4x2 + 3x3 = 2, 2x1 + 7x2 + x3 = 1,d) 2x1 3x2 + 4x3 = 5, x2 x3 = 4, x1 + 3x2 x3 = 2.

(15) Consider the following system of equations:x + 2y + z = 3, ay + 5z = 10, 2x + 7y + az = b.a) Find all values of a for which the following system of equations has a unique solution.b) Find all pairs (a, b) for which the system has more than one solution.

(16) Find a,b,c,p,q R such that the following system has a solution:

1 2 30 1 p

0 0 q

x1x2

x3

=

ab

c

.

::END::