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(MATH111)[2010](f)quiz~2247^_10438.pdf
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MATH 111 Linear Algebra Quiz 1 for T2a
Name:
Student ID: Time allowed: 20 minutes

1. Find a suitable 2 2 invertible matrix P such that PA is in reduced row echelon form.
. 1 5 3 5 .
A = 3 15 .3 3 .
2. Let A be a 2 2 matrix such that:

12 15
A = ,A = .
11 .13
Write the above as a matrix equation AB = C and solve for A. END
1. We perform row operations on [ A | I2 ]:
15 35
10 .3r1+r2 153 5
10
.

3 15 .33
01 00 .12 .12
.31
. 1
12
1
4
1
4
1 4
. 1
12
r2
1535
10 .3r2+r1 1502
.


.

. 1
12
0011

0011

1
4
So when we take P

(row-equivalent to I2, so invertible), then:
=

1
4
. 1
12
P A = . 1 0 5 0 0 1 2 1 . is in RREF.
2. Let B = . b1 b2 . = . 1 1 1 .1 . . Then:
AB = . Ab1 Ab2 . = . 2 1 5 3 . = C.

1
2
1
2
Note that B is invertible with B.1
=

, so we have:

.

. .

1173 2
25

.

AB = C . A = CB.1 =
222
=

.

1 21 2
2 .1

MATH 111 Linear Algebra Quiz 1 for T2b
Name:
Student ID: Time allowed: 20 minutes

1. Find a suitable 2 2 invertible matrix P such that PA is in reduced row echelon form.
13 .63
A = .
3 .2 .7 .2
.. 12
2. Let B = b1 b2 = . Suppose that A is a 2 2 matrix such that:
23
Ab1 = b1 + e1,Ab2 = b2 + e2.
Write the above as a matrix equation AB = C and solve for A.
END

1. We perform row operations on [ A | I2 ]:
13 .63
10 .3r1+r2 13 .63
10
.

3 .2 .7 .2
01 0 .11 11 .11
.31
. 1 r2
23
11 13 .63 10 .3r2+r1 10 .30
11 11
.
.

3 . 1
01 .11 3 . 1 01 .11
11 11 11 11
23
11 11

So when we take P = (row-equivalent to I2, so invertible), then:
3 . 1
11 11

10 .30
PA = is in RREF.
01 .11
2. We have
AB = Ab1 Ab2 = b1 + e1 b2 + e2 = B + I.
.32
Therefore we get (A . I)B = I. Since B is invertible with B.1 = , we have:
2 .1
.22
A . I = B.1 . A = B.1 + I = .
20