9. Let A be the coefficient matrix for the system of linear equations: 2x + 9y – 5z = 1, — 6х — 2у + 10z = -8, (1) 9х + 3у — 152 = 12; and matrices 1 b12 -2 1 3 В- 2 -2 C = 1 2 -2 |b31 b32 (a) (i) Given that A В С, find bi2, bsз1, bз2- (ii) Note that C transforms a generic 3-d vector x = x y z' to a vector on the (x, y)-plane given by [-2x + y + 3z] Cx = x + 2y – 2z Noting the above and that A =B C, deduce the determinant of A providing a clear geometric explanation. (b) Find the solution(s) to the system of linear equations (1). (c) (i) Show that the basis vectors [1 0 0]', [0 1 0]',[o o 1]', and the vector [1 1 1]',. transformation by A. are mapped to the plane defined by 3y + 2z = 0 through (ii) Show that there is a point on the plane 3y + 2z = 0 at x = 23/13 that is a solution to the system of linear equations (1).
9. Let A be the coefficient matrix for the system of linear equations: 2x + 9y – 5z = 1, — 6х — 2у + 10z = -8, (1) 9х + 3у — 152 = 12; and matrices 1 b12 -2 1 3 В- 2 -2 C = 1 2 -2 |b31 b32 (a) (i) Given that A В С, find bi2, bsз1, bз2- (ii) Note that C transforms a generic 3-d vector x = x y z' to a vector on the (x, y)-plane given by [-2x + y + 3z] Cx = x + 2y – 2z Noting the above and that A =B C, deduce the determinant of A providing a clear geometric explanation. (b) Find the solution(s) to the system of linear equations (1). (c) (i) Show that the basis vectors [1 0 0]', [0 1 0]',[o o 1]', and the vector [1 1 1]',. transformation by A. are mapped to the plane defined by 3y + 2z = 0 through (ii) Show that there is a point on the plane 3y + 2z = 0 at x = 23/13 that is a solution to the system of linear equations (1).
Calculus: Early Transcendentals
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Chapter1: Functions And Models
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![9. Let A be the coefficient matrix for the system of linear equations:
2х + 9у — 52
1,
— 6х — 2у + 102
:-8,
(1)
9х + 3у — 15г
= 12;
and matrices
1
b12
-2 1
3
В -
-2
С -
1
2
2
[b31 b32.
(a) (i) Given that A = BC, find b12, b31, b32 .
(ii) Note that C transforms a generic 3-d vector x = x y z' to a vector on
the (x, y)-plane given by
-2x + y + 3z
x + 2y – 2z
Cx =
Noting the above and that A = B C, deduce the determinant of A
providing a clear geometric explanation.
(b) Find the solution(s) to the system of linear equations (1).
(c) (i) Show that the basis vectors [1 0 0]', [0 1 0]', [0 o 1]', and the
vector 1 1 1', are mapped to the plane defined by 3y + 2z = 0 through
transformation by A.
(ii) Show that there is a point on the plane 3y + 2z = 0 at x =
solution to the system of linear equations (1).
23/13 that is a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ebc4e0-ed89-4bec-bb3b-3de7d1b804aa%2F7131fc1e-bf3a-4f37-8252-ae008e34d0dc%2Ffnwuvxh_processed.png&w=3840&q=75)
Transcribed Image Text:9. Let A be the coefficient matrix for the system of linear equations:
2х + 9у — 52
1,
— 6х — 2у + 102
:-8,
(1)
9х + 3у — 15г
= 12;
and matrices
1
b12
-2 1
3
В -
-2
С -
1
2
2
[b31 b32.
(a) (i) Given that A = BC, find b12, b31, b32 .
(ii) Note that C transforms a generic 3-d vector x = x y z' to a vector on
the (x, y)-plane given by
-2x + y + 3z
x + 2y – 2z
Cx =
Noting the above and that A = B C, deduce the determinant of A
providing a clear geometric explanation.
(b) Find the solution(s) to the system of linear equations (1).
(c) (i) Show that the basis vectors [1 0 0]', [0 1 0]', [0 o 1]', and the
vector 1 1 1', are mapped to the plane defined by 3y + 2z = 0 through
transformation by A.
(ii) Show that there is a point on the plane 3y + 2z = 0 at x =
solution to the system of linear equations (1).
23/13 that is a
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