A proof problem of College Linear Algebra: let n-order matrix a satisfy the square of a = a, e is n-order unit matrix, and prove that R (a) + R (A-E) = n

This is a very simple line generation proof!
Because a ^ 2 = a, a (A-E) = 0
The results are as follows:
R (a) + R (A-E) less than or equal to n
And because (A-E) + (- a) = - E
The results are as follows:
R (- a) + R (A-E) greater than or equal to n
Because R (- a) = R (a)
So r (a) + R (A-E) is greater than or equal to n
It can be seen from the pinch theorem that:
R (a) + R (A-E) equals n
Chen Wendeng's mathematics postgraduate entrance examination guidance has a special introduction, is the use of a theorem!
I believe it can solve your problem!

Matrix problem of linear algebra in University
I want to know if it is possible to find the rank of a matrix and transform it into other matrices rather than row ladder matrices. For example, a fourth-order square matrix can be transformed into another matrix (not row ladder matrix), but I can be sure that the last row is 0, and the third-order subformula is not 0
When can I use column elementary transformation? My teacher talked about it in class, but didn't listen to it carefully

Your method is OK
The elementary transformation does not change the rank of the matrix
Look at the situation
You become a row ladder matrix, and that's clear
Some don't need you to change. You can find his rank at the end

College linear algebra matrix problem solving, 1 using adjoint matrix to find inverse matrix (- 1, 2 - 3)
Using adjoint matrix to find inverse matrix-1, 2, 3
( 2 1 0 )
4 -2 5
The second problem uses the elementary transformation of matrix to calculate the inverse of the following matrix
（1） 2 1 -1 （2） 1 -2 1 0
（ 0 2 1 ）（ 0 1 -2 1 ）
5 2 -3 0 0 0 -2
0 0 0 1

1. |a| = - 1, inverse matrix = - A * = -
〔 5 -4 3〕
〔-10 7 -6〕
〔-8 6 -5〕
2、（1)
A＝
2 1 -1
0 2 1
5 2 -3 ?
A inverse matrix =
8 -1 -3
-5 1 2
10 -1 -4
(2)A＝
1 -2 1 0
0 1 -2 1
0 0 0 -2
01? Irreversible

The problem of matrix in linear algebra
A, B, C, D and e each borrowed a novel from the library. They agreed to read and exchange the five books. The thickness of the five books and their reading speed were similar. Therefore, the five people always exchanged books at the same time. After four exchanges, the five of them finished the five books
(1) The last book read by a is the second book read by B;
(2) The last book read by C is the fourth book read by B;
(3) C read the second book a at the beginning;
(4) The last book Ding read was the third one c read;
(5) The fourth book read by B is the third book read by E;
(6) The book Ding read for the third time was the one Bing read at the beginning
According to the above situation, who read the book Ding read the second time first?

Let a, B, C, D and e read the last book code in turn, then the following initial matrix can be listed according to the conditions: in the above matrix, X and Y represent the undetermined title code, and the same letter represents the same book

6.4x + 3 = 15.8 (solution equation)

6.4x=15.8-3
6.4x=12.8
x=12.8÷6.4
x=2

1.5 * 999 + 5 + 99 * 7 + 7 + 3 * 9 + 3 + 9 2.9999 * 7 + 11111 * 37 simple method

（1）=5×﹙999﹢1﹚＋7×﹙99＋1﹚＋3×﹙9＋1﹚=5000＋700＋30=5730
（2﹚=11111×9×7＋11111×37=11111×﹙63＋37﹚=1111100

Ten questions of filling in the blanks in Mathematics
Fill in the brackets with the appropriate unit of measurement
The train runs 120 per hour. A pencil weighs about 5. A tree is about 9
A watermelon weighs about 4 (). The classroom covers an area of about 42 (). Beibei's height is 130 ()
The school playground is about 1 (). The area of a handkerchief is about 4 ()
It takes 20 () for Beibei to run 100 meters. China's land area is about 9.6 million ()

The train runs 120 kilometers per hour. A pencil weighs about 5 grams. A tree is about 9 meters high
A watermelon weighs about 4 kg. The classroom covers an area of about 42 square meters. Beibei's height is 130 cm
The school playground is about 1 square kilometer. The area of a handkerchief is about 4 square decimeters
It takes Beibei 20 seconds to run 100 meters. China's land area is about 9.6 million square kilometers

The original price of a commodity was 400 yuan, and the price was reduced by 10% for two consecutive times. What's the current price

The original price of a commodity was 400 yuan, and it was sold at a 10% discount twice in a row,
Now the price is 400x0.9x0.9 = 324 yuan

The domain of the function y = lgsinx is , the range is ．
The fixed sum of cos3x with y = 2 radical

1. Domain of definition (2k π, 2K π + π), K ∈ Z
The range is (- ∞, 0]
2. Domain [2 / 3K π - π / 6,2 / 3K π + π / 6] K ∈ Z
Range [0,2]

Practical problems and quadratic equations of one variable
Based on the social and economic benefits, a three-year plan has been formulated to invest in the construction of ecological environment and develop the tourism industry. According to the plan, the investment in the first year is 8 million yuan, the second year is 1 / 3 less than the first year, the third year is 1 / 2 less than the second year, and the local tourism income in the first year is estimated to be 4 million yuan, What is the average annual growth rate of tourism revenue if the investment in three years is equal to the total tourism revenue? (the following data are available: root 2 ≈ 1.414, root 3 ≈ 3.606, the calculation result is accurate to the percentile)

Investment,
In the first year: 8 million yuan;
In the second year: 800 × (1-1 / 3) (ten thousand yuan);
The third year is: 800 × (1-1 / 3) × (1-1 / 2) ((ten thousand yuan);
The sum of the three years is 16 million yuan
If the average annual growth rate is x, then:
In the first year: 4 million yuan;
The second year is: 400 × (1 + x) (ten thousand yuan);
The third year is: 400 × (1 + x) + (1 + x) ^ 20000 yuan;
The total income of three years is: 400 + 400 (1 + x) + 400 (1 + x) ^ 2
Therefore, according to the equality of revenue and expenditure, it can be concluded that:
400+400（1+x）+400（1+x）^2=1600；
Simplification is: x ^ 2 + 3x-1 = 0;
By using the root formula of quadratic equation with one variable, it is obtained that: 1
X = ((radical 13) - 3) / 2 ≈ 0.303;
Therefore, in order to make the investment in three years equal to the total tourism revenue, the average annual growth rate of tourism revenue should be 30.3%

A proof problem of College Linear Algebra: let n-order matrix a satisfy the square of a = a, e is n-order unit matrix, and prove that R (a) + R (A-E) = n