Timothy 發問時間： 科學數學 · 6 年前

# 三題有關數學矩陣Matrices的計算

1. Let

A =3 −1 2

0 2 −1

1 1 1

Compute the determinant of A and find the inverse of A (if exists). Is A singular? Compute

A−1(A′ + 3I3).

2. Find a general solution to

x1 + 2x2 − 4x3 + x4 = 1

−2x1 + x2 − 2x3 − x4 = 2.

(use Gaussian elimination) Is the system consistent? How many degrees of freedom it possesses?

Find a particular solution (if exists). Is (0, 0, 4, −1) a solution to the system?

3. You are given the following system of linear equations:

2x1 + x2 = 1

−2x1 + 4x2 = 3.

Write down the coefficient matrix of the system and its rhs vector. Is the system Cramer’s system?

If so, then find its solution using Cramer’s rule (formulas).

### 2 個解答

• 麻辣
Lv 7
6 年前
最佳解答

1.A =3 −1 2

0 2 −1

1 1 1

1-1.Determinant of A = ?det = 6 + 1 + 0 - 4 + 3 - 0 = 6

1-2.Inverse of A (if exists) = ?b11 = |2 -1; 1 1| = 3b12 = -|0 -1; 1 1| = -1b13 = |0 2 ; 1 1| = -2

b21 = -|-1 2 ; 1 1| = 3b22 = |3 2; 1 1| = 1b23 = -|3 -1; 1 1| = -4

b31 = |-1 2; 2 -1| = -3b32 = -|3 2; 0 -1| = 3 b33 = |3 -1; 0 2| = 6

B =|.3 .1 -2|'

|-3 .1 .4|/det =

|-3 .3 .6||.3 .3 -3|

|-1 .1 .3|/6 =

|-2 -4 .6||.0.5 .0.5 -0.5|

|-1/6 .1/6 .0.5|

|-1/3 -2/3 .1..|

1-3.Is A singular?

det =\=0 = No

1-4. w = A(−1)*(A′ + 3*I*3) = ?B = A^(-1) =|.3 .3 -3|

|-1 .1 .3|/6

|-2 -4 .6|

C = A' = |.3 .0 1|

|-1 .2 1|

|.2 -1 1|

3*I*3 =|9 0 0|

|0 9 0|

|0 0 9|

w = B*C + 9*I =|.36 -9 .-36|

|-9 ..20 .16|/6 =

|-36 .16 .65| |.6 .-3/2 -6...|

|-3/2 10/3 8/3.|

|-6 ..8/3 .65/6|

2. Find a general solution to.x1 .+ 2x2 - 4x3 + x4 = 1

−2x1 + .x2 - 2x3 - x4 = 2

2-1.Use Gaussian eliminationx1 + 2x2 = 1 + 4x3 - x4-2x1 + x2 = 2 + 2x3 + x4

|.1 2 1+4x3-x4|--a

|-2 1 2+2x3+x4|--b保留a: 2b-a|.1 2 1+4x3-x4|

|-5 0 3+3x4___|

-5x1 = 3 + 3x4 ==> x1 = -3(1+x4)/5x2 = (1+4x3-x4-x1)/2= [5(1+4x3-x4)+(3+3x4)]/10= (5 + 20x3 - 5x4 + 3 + 3x4)/10= (20x3 - 2x4 + 8)/10= (10x3 - x4 + 4)/5

2-2.Is the system consistent? Ans: No!!Variables = {x1 x2; x1 x3; x1 x4; x2 x3; x2 x4; x3 x4}

2-3.How many degrees of freedom it possesses?Ans: Variables = 2 ==> dof = 2

2-4.Find a particular solution if exists. (x3 x4) = (1 -6)x1 = -3(1+x4)/5= 3*5/5= 3

x2 = (10x3 - x4 + 4)/5= (10 + 6 + 4)/5= 4

2-5.Is (0, 0, 4, −1) a solution to the system?Ans: No!!

x2 = (10x3 - x4 + 4)/5= (40 + 1 + 4)/5= 45/5= 9

3. Given the following system of linear equations:.2x1 + .x2 = 1

−2x1 + 4x2 = 3

3-1.Write down the coefficient matrix of the system [.2 1]{x1}.{1}

[-2 4]{x2}={3}

3-2.And its rhs vector.

A*X = BA = [2 1; -2 4]X = {x1 x2}'B = {1 3}'

3-3.Is the system Cramer's system?Ans: Yes!!

2015-06-03 09:36:30 補充：

3-4.Then find its solution using Cramer's rule (formulas).

Ans:

(x1, x2) = Simple form for Cramer

=|.2 1 -1 .2|

.|-2 4 -3 -2|

= (2*4+2*1; -1*3+4*1; 1*2+3*2)

= (10; 1; 8)

= (1/10, 8/10)

= (0.1, 0.8)

• 6 年前

不好意思 想請問您是否知道 哪個網站或哪本書 對矩陣有 有效率的解說或教學 我個人認為我對矩陣的基本觀念 還不是很清楚

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