MATH 152 Midterm 2
Review Session March 18, 2015.
SOLUTIONS
⎛ 4 2 0 ⎞
⎛ 2 3 1 ⎞
⎛ 3 1 −3 ⎞
⎜
⎟
⎜
⎟
⎜
⎟
1) Find AB and AC if A = ⎜ 2 1 0 ⎟ , B = ⎜ 2 −2 −2 ⎟ , C = ⎜ 0 2 6 ⎟
⎝ −2 −1 1 ⎠
⎝ −1 2 1 ⎠
⎝ −1 2 1 ⎠
4(3) + 2(−2) + 0(2)
4(1) + 2(−2) + 0(1)
⎛ 4 2 0 ⎞ ⎛ 2 3 1 ⎞ ⎛ 4(2) + 2(2) + 0(−1)
⎜
⎜
⎟
⎜
⎟
AB =
2(2) + 1(2) + 0(−1)
2(3) + 1(−2) + 0(2)
2(1) + 1(−2) + 0(1)
2 1 0
2 −2 −2 = ⎜
⎜
⎟⎜
⎟
⎝ −2 −1 1 ⎠ ⎝ −1 2 1 ⎠ ⎜⎝ −2(2) + (−1)(2) + 1(−1) −2(3) + (−1)(−2) + 1(2) −2(1) + (−1)(−2) + 1(1)
⎞
⎟
⎟
⎟⎠
⎛ 12 8 0 ⎞
=⎜ 6 4 0 ⎟
⎜
⎟
⎝ −7 −2 1 ⎠
4(1) + 2(2) + 0(2)
4(−3) + 2(6) + 0(1)
⎛ 4 2 0 ⎞ ⎛ 3 1 −3 ⎞ ⎛ 4(3) + 2(0) + 0(−1)
⎜
AC = ⎜ 2 1 0 ⎟ ⎜ 0 2 6 ⎟ = ⎜
2(3) + 1(0) + 0(−1)
2(1) + 1(2) + 0(2)
2(−3) + 1(6) + 0(1)
⎜
⎟⎜
⎟
⎝ −2 −1 1 ⎠ ⎝ −1 2 1 ⎠ ⎜⎝ −2(3) + (−1)(0) + 1(−1) −2(1) + (−1)(2) + 1(2) −2(−3) + (−1)(6) + 1(1)
⎞
⎟
⎟
⎟⎠
⎛ 12 8 0 ⎞
=⎜ 6 4 0 ⎟
⎜
⎟
⎝ −7 −2 1 ⎠
NOTE that AC = AB, but B≠C. This shows that the cancellation law is not valid for matrices. Also note that A is not invertible,
so one could NOT multiply both sides of AC = AB by A inverse to obtain B = C.
2) Find the determinant of the matrix
⎛
⎜
⎜
⎜
⎜
⎜
⎜⎝
2 6 log 2
0
0
0
0
5
0
0
0
2
π
0
0
π2
4
sin(9)
−4
0
e ⎞
⎟
5 ⎟
7 ⎟
⎟
21 ⎟
6 ⎟⎠
2 6 log 2
0
0
0
0
5
0
0
0
2
π
0
0
π2
4
sin(9)
−4
0
e
5
7 = (2)(5)(π )(−4)(6) = −240π
21
6
3) What matrix has the effect of rotating every vector through 90 degrees counter clockwise and
then projecting the result onto the x-axis?
⎛ cos(90) − sin(90) ⎞ ⎛ 0 −1 ⎞
R=⎜
⎟ =⎜
⎟
⎝ sin(90) cos(90) ⎠ ⎝ 1 0 ⎠
⎛ 1 0 ⎞
⎛ 1 0 ⎞ ⎛ 0 −1 ⎞ ⎛ 0 −1 ⎞
P=⎜
→ PR = ⎜
=
⎟
⎝ 0 0 ⎠
⎝ 0 0 ⎟⎠ ⎜⎝ 1 0 ⎟⎠ ⎜⎝ 0 0 ⎟⎠
4) What matrix represents projection onto the x-axis followed by projection onto the y-axis?
⎛ 0 0 ⎞
⎜⎝ 0 0 ⎟⎠ the zero matrix performs this operation, because every vector will be
mapped into the zero vector. Projecting a vector onto the x-axis will remove its y-componenent,
therefore when projecting onto the y-axis, there is no y-component left, and you get the zero vector.
5) If
⎛ ⎡ 2 ⎤⎞ ⎡ 5 ⎤
⎛ ⎡1 ⎤⎞ ⎡ −2 ⎤
⎛ ⎡1 ⎤⎞
T : ! 2 → ! 2 , T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ , T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ , find a) T ⎜ ⎢ ⎥⎟ ,
⎝ ⎣ 3 ⎦⎠ ⎣ 7 ⎦
⎝ ⎣ 4 ⎦⎠ ⎣ 3 ⎦
⎝ ⎣ −1⎦⎠
b) find the matrix for the linear transformation T,
c) find the inverse transformation T −1 : ! 2 → ! 2
⎛ ⎡ 2 ⎤⎞ ⎡ 5 ⎤
⎛ ⎡1 ⎤⎞ ⎡ −2 ⎤
⎛ ⎡1 ⎤⎞
T : ! 2 → ! 2 , T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ , T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ , find a) T ⎜ ⎢ ⎥⎟ ,
⎝ ⎣ 3 ⎦⎠ ⎣ 7 ⎦
⎝ ⎣ 4 ⎦⎠ ⎣ 3 ⎦
⎝ ⎣ −1⎦⎠
⎛ ⎡1 ⎤⎞
⎛ ⎡ 2 − 1 ⎤⎞
⎛ ⎡ 2 ⎤⎞
⎛ ⎡1 ⎤⎞ ⎡ 5 ⎤ ⎡ −2 ⎤ ⎡ 7 ⎤
T ⎜ ⎢ ⎥⎟ = T ⎜ ⎢
= T ⎜ ⎢ ⎥⎟ − T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥
⎥
⎟
⎝ ⎣ −1⎦⎠
⎝ ⎣ 3 − 4 ⎦⎠
⎝ ⎣ 3 ⎦⎠
⎝ ⎣ 4 ⎦⎠ ⎣ 7 ⎦ ⎣ 3 ⎦ ⎣ 4 ⎦
b) find the matrix for the linear transformation T,
⎛ ⎡ 2 ⎤⎞
⎛ ⎡1 ⎤⎞
⎛ ⎡ 0 ⎤⎞ ⎡ 5 ⎤
⎛ ⎡ 0 ⎤⎞
⎛ ⎡ 0 ⎤⎞ ⎡ −9 / 5 ⎤
⎡ −2 ⎤ ⎡ 9 ⎤
T ⎜ ⎢ ⎥⎟ − 2T ⎜ ⎢ ⎥⎟ = T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ − 2 ⎢ ⎥ = ⎢ ⎥ = −5T ⎜ ⎢ ⎥⎟ → T ⎜ ⎢ ⎥⎟ = ⎢
⎥
⎝ ⎣ 3 ⎦⎠
⎝ ⎣ 4 ⎦⎠
⎝ ⎣ −5 ⎦⎠ ⎣ 7 ⎦
⎝ ⎣1 ⎦⎠
⎝ ⎣1 ⎦⎠ ⎣ −1 / 5 ⎦
⎣ 3 ⎦ ⎣1 ⎦
⎛ ⎡1 ⎤⎞
⎛ ⎡1 ⎤⎞
⎛ ⎡ 0 ⎤⎞ ⎡ 7 ⎤ ⎡ −9 / 5 ⎤ ⎡ 26 / 5 ⎤
⎛ ⎡1 ⎤⎞ ⎡ 26 / 5 ⎤
T ⎜ ⎢ ⎥⎟ = T ⎜ ⎢ ⎥⎟ + T ⎜ ⎢ ⎥⎟ = ⎢ ⎥ + ⎢
=⎢
→ T ⎜ ⎢ ⎥⎟ = ⎢
⎥
⎥
⎥
⎝ ⎣ 0 ⎦⎠
⎝ ⎣ −1⎦⎠
⎝ ⎣1 ⎦⎠ ⎣ 4 ⎦ ⎣ −1 / 5 ⎦ ⎣19 / 5 ⎦
⎝ ⎣ 0 ⎦⎠ ⎣19 / 5 ⎦
⎛ 26 / 5 −9 / 5 ⎞ 1 ⎛ 26 −9 ⎞
Matrix = ⎜
=
⎝ 19 / 5 −1 / 5 ⎟⎠ 5 ⎜⎝ 19 −1 ⎟⎠
c) find the inverse transformation T −1 : ! 2 → ! 2
⎛ 26 / 5 −9 / 5 ⎞
⎜⎝ 19 / 5 −1 / 5 ⎟⎠
−1
=
1 ⎛ −1 9 ⎞
29 ⎜⎝ −19 26 ⎟⎠
6) Compute the product
⎛ 3 −6 0 ⎞ ⎛ 2 ⎞
!
Ax = ⎜ 0 2 −2 ⎟ ⎜ 1 ⎟
⎜
⎟⎜ ⎟
⎝ 1 −1 −1 ⎠ ⎝ 1 ⎠
For this matrix A, find a solution vector x to the system Ax = 0, with zeroes on the right side of
all three equations. Can you find more than one solution?
⎛ 3 −6 0 ⎞ ⎛ 2 ⎞ ⎛ 3(2) + (−6)(1) + 0(1) ⎞ ⎛ 0 ⎞
!
Ax = ⎜ 0 2 −2 ⎟ ⎜ 1 ⎟ = ⎜ 0(2) + 2(1) + (−2)(1) ⎟ = ⎜ 0 ⎟
⎟ ⎜ ⎟
⎜
⎟⎜ ⎟ ⎜
⎝ 1 −1 −1 ⎠ ⎝ 1 ⎠ ⎝ 1(2) + (−1)(1) + (−1)(1)⎠ ⎝ 0 ⎠
⎛ 2⎞
any multiple of ⎜ 1 ⎟ would also produce the zero vector.
⎜ ⎟
⎝1 ⎠
7) Find the inverse of the matrix A.
⎛ 1 −2 3 ⎞
A=⎜ 3 5 1 ⎟
⎜
⎟
⎝ 6 4 2 ⎠
⎛ 1 −2 3 1 0 0 ⎞ ⎛ 1 −2
⎛ 1 −2 3 ⎞
3
1 0 0 ⎞ ⎛ 1 −2 3
1
0
0
A = ⎜ 3 5 1 ⎟ → ( A | I ) = ⎜ 3 5 1 0 1 0 ⎟ → ⎜ 0 11 −8 −3 1 0 ⎟ → ⎜ 0 11 −8
−3
1
0
⎜
⎟ ⎜
⎜
⎟
⎟ ⎜
6
4
2
0
0
1
⎝
⎠ ⎝ 0 16 −16 −6 0 1 ⎠ ⎝ 0 1 −1 −6 /16 0 1 /16
⎝ 6 4 2 ⎠
⎛ 1 −2 3
1
0
0
→ ⎜ 0 1 −1 −6 /16 0 1 /16
⎜
−3
1
0
⎝ 0 11 −8
⎞ ⎛ 1 −2 3
1
0
0
⎟ → ⎜ 0 1 −1
−3 / 8
0 1 /16
⎟ ⎜
⎠ ⎝ 0 0 3 −3 + 33 / 8 1 −11 /16
⎛ 1 −2 0 −1 / 8 −1 11 /16
→⎜ 0 1 0
0
1 / 3 −1 / 6
⎜
⎝ 0 0 1 3 / 8 1 / 3 −11 / 48
⎛ −1 / 8 −1 / 3 17 / 48
A −1 = ⎜ 0
1/ 3
−1 / 6
⎜
3
/
8
1
/
3
−11
/ 48
⎝
⎞
⎟ = ( I | A −1 )
⎟
⎠
⎞
⎛ −6 −16 17 ⎞
⎟ = 1 ⎜ 0 16 −8 ⎟
⎟ 48 ⎜
⎟
⎠
⎝ 18 16 −11 ⎠
8) Determine the inverse of
⎛ 1 2 3
A=⎜ 4 5 6
⎜
⎝ 7 8 9
⎞ ⎛ 1 0 0 −1 / 8 −1 / 3 −17 / 48
⎟ →⎜ 0 1 0
0
1/ 3
−1 / 6
⎟ ⎜
⎠ ⎝ 0 0 1 3 / 8 1 / 3 −11 / 48
⎞ ⎛ 1 −2 3
1
0
0
⎟ → ⎜ 0 1 −1 −3 / 8 0
1 /16
⎟ ⎜
⎠ ⎝ 0 0 1 3 / 8 1 / 3 −11 / 48
⎛ 1 2 3
A=⎜ 4 5 6
⎜
⎝ 7 8 9
⎞
⎟
⎟
⎠
⎞
⎛ 1 2 3 1 0 0
⎟ → (A | I ) = ⎜ 4 5 6 0 1 0
⎟
⎜
⎠
⎝ 7 8 9 0 0 1
⎞ ⎛ 1 2
3
1 0 0 ⎞
⎟ → ⎜ 0 −3 −6 −4 1 0 ⎟
⎟ ⎜
⎟
⎠ ⎝ 0 −6 −12 −7 0 1 ⎠
note that the last two rows of A after the first elimination step are multiples of each other.
Therefore A is NOT invertible. Check that its determinant is zero.
⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎠
9) Show that the inverse of A is equal to the transpose of A.
⎛ 0 1 0
A=⎜ 1 0 0
⎜
⎝ 0 0 1
⎞
⎛ 0 1 0 1 0 0
⎟ → (A | I ) = ⎜ 1 0 0 0 1 0
⎟
⎜
⎠
⎝ 0 0 1 0 0 1
⎛ 0 1 0 ⎞
A=⎜ 1 0 0 ⎟
⎜
⎟
⎝ 0 0 1 ⎠
⎞ ⎛ 1 0 0 0 1 0 ⎞
⎟ → ⎜ 0 1 0 1 0 0 ⎟ = ( I | A −1 )
⎟ ⎜
⎟
⎠ ⎝ 0 0 1 0 0 1 ⎠
⎛ 0 1 0 ⎞
A = ⎜ 1 0 0 ⎟ = A −1 = A
⎜
⎟
⎝ 0 0 1 ⎠
T
10) Find the transpose of
⎛
⎜
A=⎜
⎜
⎜
⎝
⎛
A=⎜
⎜
⎜
⎜
⎝
−6 9 0 ⎞
1 −1 0 ⎟
⎟
2 π 3 ⎟
5 2 6 ⎟⎠
−6 9 0 ⎞
⎛ −6 1 2 5 ⎞
1 −1 0 ⎟
T
⎟ → A = ⎜ 9 −1 π 2 ⎟
2 π 3 ⎟
⎜
⎟
⎝ 0 0 3 6 ⎠
⎟
5 2 6 ⎠
11) Compute AB and BA if
⎛ 1 2 3 ⎞
⎛ 7 8 ⎞
A=⎜
,
B
=
⎜⎝ 0 −9 ⎟⎠
⎝ 4 −5 6 ⎟⎠
⎛ 1 2 3 ⎞
⎛ 7 8 ⎞
A=⎜
, B=⎜
⎟
⎝ 4 −5 6 ⎠
⎝ 0 −9 ⎟⎠
⎛ 1 2 3 ⎞⎛ 7 8 ⎞
AB = ⎜
= UNDEFINED
⎝ 4 −5 6 ⎟⎠ ⎜⎝ 0 −9 ⎟⎠
⎛ 7 8 ⎞ ⎛ 1 2 3 ⎞ ⎛ 7(1) + 8(4)
7(2) + 8(−5)
7(3) + 8(6) ⎞
BA = ⎜
=
⎜
⎟
⎝ 0 −9 ⎟⎠ ⎜⎝ 4 −5 6 ⎟⎠ ⎝ 0(1) + (−9)(4) 0(2) + (−9)(−5) 0(3) + (−9)(6) ⎠
⎛ 39 −26 69 ⎞
BA = ⎜
⎝ −36 45 −54 ⎟⎠
! ⎛ −1 3 ⎞ !
! ⎛ 3 −2 6 ⎞ !
T x =⎜
x
,
S(x
)=⎜
x
⎝ 9 4 ⎟⎠
⎝ −4 6 2 ⎟⎠
()
12) If
find
⎛ ⎡ 1 ⎤⎞
⎛ ⎡ 2 ⎤⎞
⎛ ⎡ −2 ⎤⎞
⎥⎟
⎜⎢
T ! S, S !T , T ⎜ ⎢
⎥⎟ , S ⎜ ⎢
⎥⎟ , S ⎜ ⎢ 2 ⎥⎟
⎝ ⎣ 1 ⎦⎠
⎝ ⎣ 4 ⎦⎠
⎜⎝ ⎣⎢ 3 ⎥⎦⎟⎠
⎛ −1 3 ⎞ ⎛ 3 −2 6 ⎞ ⎛ −1(3) + 3(−4) −1(−2) + 3(6) −1(6) + 3(2) ⎞ ⎛ −15 20 0 ⎞
T !S = ⎜
=⎜
⎟=
⎝ 9 4 ⎟⎠ ⎜⎝ −4 6 2 ⎟⎠ ⎝ 9(3) + 4(−4) 9(−2) + 4(6) 9(6) + 4(2) ⎠ ⎜⎝ 11 6 62 ⎟⎠
S !T = UNDEFINED
⎛ ⎡ 2 ⎤⎞ ⎛ −1 3 ⎞ ⎛ 2 ⎞ ⎛ −1(2) + 3(1)⎞ ⎛ 1 ⎞
T ⎜ ⎢ ⎥⎟ = ⎜
=
=
⎝ ⎣1 ⎦⎠ ⎝ 9 4 ⎟⎠ ⎜⎝ 1 ⎟⎠ ⎜⎝ 9(2) + 4(1) ⎟⎠ ⎜⎝ 22 ⎟⎠
⎛ ⎡ −2 ⎤⎞
S ⎜ ⎢ ⎥⎟ = UNDEFINED
⎝ ⎣ 4 ⎦⎠
⎛ ⎡1 ⎤⎞
⎛1 ⎞
⎛
⎞
3
−2
6
⎜ 2 ⎟ = ⎛ 3(1) + (−2)(2) + 6(3)⎞ = ⎛ 17 ⎞
S ⎜ ⎢ 2 ⎥⎟ = ⎜
⎟ ⎜ ⎟
⎟
⎢
⎥
⎜
⎟
⎜ ⎟ ⎜
⎜⎝ ⎢ 3 ⎥⎟⎠ ⎝ −4 6 2 ⎠ ⎝ 3⎠ ⎝ −4(1) + 6(2) + 2(3) ⎠ ⎝ 14 ⎠
⎣ ⎦
13) Find the matrix R that rotates a vector by 225 degrees counterclockwise, and then find the
image of < 2, 5 > under this linear transformation.
⎛ cosθ
R=⎜
⎝ sin θ
R=
⎛ cos(5π / 4) − sin(5π / 4) ⎞ ⎛ −1 / 2
− sin θ ⎞
,
θ
=
5
π
/
4
→
R
=
⎜
⎟ = ⎜⎜
cosθ ⎟⎠
sin(5
π
/
4)
cos(5
π
/
4)
⎝
⎠ ⎝ −1 / 2
⎛ ⎡ 2 ⎤⎞ −1 ⎛ 1 −1 ⎞ ⎛ 2 ⎞ −1 ⎛ 1(2) + (−1)(5)⎞ −1 ⎛ −3⎞
−1 ⎛ 1 −1 ⎞
→ R ⎜ ⎢ ⎥⎟ =
=
=
⎜
⎟
2⎝ 1 1 ⎠
2 ⎜⎝ 1 1 ⎟⎠ ⎜⎝ 5 ⎟⎠
2 ⎜⎝ 1(2) + 1(5) ⎟⎠
2 ⎜⎝ 7 ⎟⎠
⎝ ⎣ 5 ⎦⎠
1/ 2 ⎞
⎟
−1 / 2 ⎟⎠
14) Find the matrix R that reflects vectors across the line y = -2x.
Show that R² = I.
Reflect the vector < -2, 3> across the line y = -2x.
!
⎡ −1⎤
x = direction of line = ⎢ ⎥
⎣2 ⎦
⎛ 1 −2 ⎞
⎡ −1⎤
⎡⎣ −1 2 ⎤⎦
! !T
⎢
⎥
⎜⎝ −2 4 ⎟⎠ ⎛
⎛ 1 0 ⎞
2
xx
1 0 ⎞ ⎛ 2 / 5 −4 / 5 ⎞ ⎛ 1 0 ⎞
R = 2 !T ! − I = ⎣ ⎦
−⎜
=
2
−⎜
=
−
⎟
5
⎡ −1⎤ ⎝ 0 1 ⎠
⎝ 0 1 ⎟⎠ ⎜⎝ −4 / 5 8 / 5 ⎟⎠ ⎜⎝ 0 1 ⎟⎠
x x
⎡⎣ −1 2 ⎤⎦ ⎢ ⎥
⎣2 ⎦
⎛ −3 / 5 −4 / 5 ⎞
R=⎜
(also note that R T = R)
⎝ −4 / 5 3 / 5 ⎟⎠
⎛ −3 / 5 −4 / 5 ⎞ ⎛ −3 / 5 −4 / 5 ⎞ 1 ⎛ 3 4 ⎞ ⎛ 3 4 ⎞ 1 ⎛ 3(3) + 4(4)
3(4) + 4(−3) ⎞
R2 = ⎜
=
=
⎜
⎟
⎝ −4 / 5 3 / 5 ⎟⎠ ⎜⎝ −4 / 5 3 / 5 ⎟⎠ 25 ⎜⎝ 4 −3 ⎟⎠ ⎜⎝ 4 −3 ⎟⎠ 25 ⎝ 4(3) + (−3)(4) 4(4) + (−3)(−3) ⎠
=
1 ⎛ 25 0 ⎞ ⎛ 1 0 ⎞
=
this is because if you reflect twice, you don't change anything overall
25 ⎜⎝ 0 25 ⎟⎠ ⎜⎝ 0 1 ⎟⎠
⎛ ⎡ −2 ⎤⎞ ⎛ −3 / 5 −4 / 5 ⎞ ⎛ −2 ⎞ 1 ⎛ −3 −4 ⎞ ⎛ −2 ⎞ 1 ⎛ −3(−2) + (−4)(3)⎞ 1 ⎛ −6 ⎞
R ⎜ ⎢ ⎥⎟ = ⎜
=
=
=
⎝ ⎣ 3 ⎦⎠ ⎝ −4 / 5 3 / 5 ⎟⎠ ⎜⎝ 3 ⎟⎠ 5 ⎜⎝ −3 3 ⎟⎠ ⎜⎝ 3 ⎟⎠ 5 ⎜⎝ −3(−2) + 3(3) ⎟⎠ 5 ⎜⎝ 15 ⎟⎠
15) Set up the augmented matrix A with the loop currents in the first three columns, and the
potential difference in the fourth column that one would rref(A) in order to solve the system.
−6 + 1(i1 ) + E = 0 → i1 + E = 6
−E + 2(i2 ) + 4(i2 − i3 ) + 7 + 3(i2 ) = 0 → 9i2 − 4i3 − E = −7
4(i3 − i2 ) + 5(i3 ) + 8 = 0 → −4i2 + 9i3 = −8
i2 − i1 = 9 → −i1 + i2 = 9
⎛
⎜
⎜
⎜
⎜⎝
1 0
0 1 6 ⎞
0 9 −4 −1 −7 ⎟
⎟
0 −4 9 0 −8 ⎟
−1 1
0 0 9 ⎟⎠
16) If each year, 1/10 of electrical engineering students transfer to computer engineering, and
2/10 of computer engineering students transfer to electrical engineering, and there are initially
400 people in electrical engineering, and 600 people in computer engineering, find the transition
matrix P, and find how many students there are in each discipline after 2 years. ⎡ 9 /10 2 /10 ⎤ ⎡ E ⎤ ⎡ 400 ⎤
⎡E ⎤
⎡E ⎤
=
P
→
P
=
⎢
⎥ , ⎢C ⎥ = ⎢ 600 ⎥
⎢C ⎥
⎢C ⎥
1
/10
8
/10
⎣ ⎦ k+1
⎣ ⎦k
⎦
⎣
⎦ ⎣ ⎦0 ⎣
⎡E ⎤
⎡ E ⎤ ⎡ 9 /10 2 /10 ⎤ ⎡ 400 ⎤ ⎡ 480 ⎤
=
P
⎢C ⎥
⎢C ⎥ = ⎢ 1 /10 8 /10 ⎥ ⎢ 600 ⎥ = ⎢ 520 ⎥
⎣ ⎦1
⎣ ⎦0 ⎣
⎦ ⎣
⎦
⎦⎣
⎡E ⎤
⎡ E ⎤ ⎡ 9 /10 2 /10 ⎤ ⎡ 480 ⎤ ⎡ 536 ⎤
=
P
⎢C ⎥
⎢C ⎥ = ⎢ 1 /10 8 /10 ⎥ ⎢ 520 ⎥ = ⎢ 464 ⎥
⎣ ⎦2
⎣ ⎦1 ⎣
⎦ ⎣
⎦
⎦⎣
17) A physics 158 course is taught in two sections, if every week 1/4 of those in section 201 and
1/3 of those in section 203 permanently drop the course, and 1/6 of each section transfer to the
other section, find the transition matrix P and the number of students in each state after 2 weeks
if initially 400 students are in section 201, and 350 students are in section 203.
⎡ 7 /12 1 / 6 0 ⎤ ⎡ S201 ⎤ ⎡ 400 ⎤
⎡ S201 ⎤
⎡ S201 ⎤
⎢ S ⎥ = P ⎢ S ⎥ → P = ⎢ 1 / 6 1 / 2 0 ⎥ , ⎢ S ⎥ = ⎢ 350 ⎥
⎢
⎥ ⎢ 203 ⎥ ⎢
⎢ 203 ⎥
⎢ 203 ⎥
⎥
1
/
4
1
/
3
1
⎢⎣
⎥⎦ ⎢⎣ D ⎥⎦ 0 ⎢⎣ 0 ⎥⎦
⎢⎣ D ⎥⎦ k+1
⎢⎣ D ⎥⎦ k
⎡ S201 ⎤
⎡ S201 ⎤ ⎡ 7 /12 1 / 6 0 ⎤ ⎡ 400 ⎤ ⎡ 875 / 3 ⎤
⎢S ⎥ = P ⎢S ⎥ = ⎢
⎥ ⎢ 350 ⎥ = ⎢ 725 / 3⎥
1
/
6
1
/
2
0
203
203
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥
⎥⎢
1
/
4
1
/
3
1
⎢⎣ D ⎥⎦1
⎢⎣ D ⎥⎦ 0 ⎢⎣
⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣ 650 / 3 ⎥⎦
⎡ S201 ⎤
⎡ S201 ⎤ ⎡ 7 /12 1 / 6 0 ⎤ ⎡ 875 / 3 ⎤ ⎡ 2525 /12 ⎤
⎢S ⎥ = P ⎢S ⎥ = ⎢
⎥ ⎢ 725 / 3⎥ = ⎢1525 / 9 ⎥
1
/
6
1
/
2
0
203
203
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥
⎥⎢
⎢⎣ D ⎥⎦ 2
⎢⎣ D ⎥⎦1 ⎢⎣ 1 / 4 1 / 3 1 ⎥⎦ ⎢⎣ 650 / 3 ⎥⎦ ⎢⎣ 5525 / 36 ⎥⎦
18) How are det(2A), det(-A), and det(A²) related to det(A), where A is an n by n matrix?
det(2A) = 2ⁿ det(A)
det(-A) = (-1)ⁿ det(A)
det(A²) = (det(A))²
19) Express
a + a' b + b'
c
d
as the sum of two other determinants.
a + a' b + b'
a b
a' b'
=
+
c
d
c d
c d
The following property of determinants always holds.
20) Show that
a − lc b − ld
a b
=
c
d
c d
a − lc b − ld
a b
c d
a b
a b
a b
=
−l
=
− l *0 =
−0=
c
d
c d
c d
c d
c d
c d
21) Evaluate det A by reducing the matrix to triangular form.
Then, find the determinants of B,C, AB, AT A,C T
⎛ 1 1 3
A=⎜ 0 4 6
⎜
⎝ 1 5 8
⎞
⎛ 1 1 3 ⎞
⎛ 1 1 3
⎟, B=⎜ 0 4 6 ⎟, C =⎜ 0 4 6
⎟
⎜
⎟
⎜
⎠
⎝ 0 0 1 ⎠
⎝ 1 5 9
⎛ 1 1 3
A=⎜ 0 4 6
⎜
⎝ 1 5 8
⎞ ⎛ 1 1 3 ⎞ ⎛ 1 1 3
⎟ →⎜ 0 4 6 ⎟ →⎜ 0 4 6
⎟ ⎜
⎟ ⎜
⎠ ⎝ 0 4 5 ⎠ ⎝ 0 0 −1
det(B) = (1)(4)(1) = 4
1 1 3
1 1 3
det(C) = 0 4 6 + 0 4 6 = −4 + 4 = 0
1 5 8
0 0 1
det(AB) = (−4)(4) = −16
det(AT ) = det(A) = −4
det(C T ) = det(C) = 0
⎞
⎟
⎟
⎠
⎞
⎟ → det(A) = (1)(4)(−1) = −4
⎟
⎠