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Educational Level Secondary school, High school and College
Program Purpose Provide step by step solutions of your problems using online calculators (online solvers)
Problem Source Your textbook, etc

1. Matrix operations
1. Addition of two matrix
2. Multiplication of two matrix
3. Power of a matrix
4. Transpose of a matrix
5. Deteminant of a matrix
6. Adjoint of a matrix
7. Inverse of a matrix
8. Prove that any two matrix expression is equal or not
9. Minor of a matrix
10. Cofactor of a matrix
11. Trace of a matrix

12. Reduce matrix
13. Rank of matrix
14. Characteristic polynomial
15. Eigen value
16. Eigen vector

2. Solve linear equation of any number of variables using
1. Inverse Matrix method
2. Cramer's Rule method
3. Gauss Elimination (Jordan) method
4. Gauss Elimination (Back Substitution) method
5. Gauss Sield method
6. Gauss Jacobi method

3. Find inverse of a matrix using
1. Simple inverse matrix
2. Gauss elimination (method of reduction)
1. Addition of two matrix
1. A + B
2. A - B
3. B - A
4. A + B + C
5. m A
6. n B
7. m A + n B

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û
 , B = 
é 3  2  1 ù
ê 1  2  1 ú
ë 2  3  0 û

Find Matrix `A + B` ...


`A + B = [[1,2,3],[0,1,0],[2,3,1]] + [[3,2,1],[1,2,1],[2,3,0]] = [[4,4,4],[1,3,1],[4,6,1]]`
2. Multiplication of two matrix
1. A × B
2. B × A
3. m × A
4. n × B
5. I × A
6. Am × Bn
7. Bm × An

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û
 , B = 
é 3  2  1 ù
ê 1  2  1 ú
ë 2  3  0 û

Find Matrix `A × B` ...


`A×B=[[1,2,3],[0,1,0],[2,3,1]]×[[3,2,1],[1,2,1],[2,3,0]]`

`=[[1*3 + 2*1 + 3*2,1*2 + 2*2 + 3*3,1*1 + 2*1 + 3*0],[0*3 + 1*1 + 0*2,0*2 + 1*2 + 0*3,0*1 + 1*1 + 0*0],[2*3 + 3*1 + 1*2,2*2 + 3*2 + 1*3,2*1 + 3*1 + 1*0]]`

`=[[11,15,3],[1,2,1],[11,13,5]]`
 
3. Power of a matrix
1. A2
2. A3
3. Am
4. A2 × A
5. Am × A
6. Am × An

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `A^2` ...


`A^2=[[1,2,3],[0,1,0],[2,3,1]]^2=[[7,13,6],[0,1,0],[4,10,7]]`
4. Transpose of a matrix
1. A'
2. B'
3. (A × B)'
4. B' × A'
5. (B × A)'
6. A' × B'
7. A + A'
8. A - A'
9. (A + B)'
10. A' + B'

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `A'` ...


`A^T=[[1,2,3],[0,1,0],[2,3,1]]^T=[[1,0,2],[2,1,3],[3,0,1]]`
 
5. Deteminant of a matrix
1. | A |
2. | B |
3. | Am |
4. | Bm |
5. | A × B |
6. | B × A |
7. | Am × Bn |
8. | Bm × An |
9. | A' |
10. | A -1 |

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `| A |` ...


`| A |=|[1,2,3],[0,1,0],[2,3,1]|`

` = 1 (1 × 1 - 0 × 3) - 2 (0 × 1 - 0 × 2) + 3 (0 × 3 - 1 × 2)`

` = 1 (1 - 0) - 2 (0 - 0) + 3 (0 - 2)`

` = 1 (1) - 2 (0) + 3 (-2)`

` = 1 - 0 - 6`

` = -5`
6. Adjoint of a matrix
1. Adj(A)
2. Adj(B)
3. Adj(Am)
4. Adj(Bm)
5. Adj(A × B)
6. Adj(B × A)
7. Adj(Am × Bn)
8. Adj(Bm × An)
9. Adj(A')
10. Adj(A -1)

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `Adj(A)` ...


`Adj(A)=Adj[[1,2,3],[0,1,0],[2,3,1]]`

`=[[+(1 × 1 - 0 × 3),-(0 × 1 - 0 × 2),+(0 × 3 - 1 × 2)],[-(2 × 1 - 3 × 3),+(1 × 1 - 3 × 2),-(1 × 3 - 2 × 2)],[+(2 × 0 - 3 × 1),-(1 × 0 - 3 × 0),+(1 × 1 - 2 × 0)]]^T`

`=[[1,0,-2],[7,-5,1],[-3,0,1]]^T`

`=[[1,7,-3],[0,-5,0],[-2,1,1]]`
 
7. Inverse of a matrix
1. A -1
2. B -1
3. (Am) -1
4. (Bm) -1
5. (A × B) -1
6. (B × A) -1
7. (Am × Bn) -1
8. (Bm × An) -1
9. (A') -1
10. (A -1) -1

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `Adj(A)` ...


`Adj(A)=Adj[[1,2,3],[0,1,0],[2,3,1]]`

`=[[+(1 × 1 - 0 × 3),-(0 × 1 - 0 × 2),+(0 × 3 - 1 × 2)],[-(2 × 1 - 3 × 3),+(1 × 1 - 3 × 2),-(1 × 3 - 2 × 2)],[+(2 × 0 - 3 × 1),-(1 × 0 - 3 × 0),+(1 × 1 - 2 × 0)]]^T`

`=[[1,0,-2],[7,-5,1],[-3,0,1]]^T`

`=[[1,7,-3],[0,-5,0],[-2,1,1]]`
8. Prove that any two matrix expression is equal or not
1. A(BC) = (AB)C
2. A(B + C) = AB + AC
3. A2 - B2 = (A - B)(A + B)
4. (AB)' = B'A'
5. (AB) -1 = (B) -1 (A) -1
6. Adj(AB) = Adj(B) Adj(A)
7. A × Adj(A) = | A | × I
8. Adj(A') = Adj(A)'
9. | A -1 | = | A | -1
10. (A') -1 = (A') -1

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û
 , B = 
é 3  2  1 ù
ê 1  2  1 ú
ë 2  3  0 û
 , C = 
é 5  2  3 ù
ê 0  3  5 ú
ë 2  0  7 û

Find Matrix `A(BC) = (AB)C` ...


`B×C=[[3,2,1],[1,2,1],[2,3,0]]×[[5,2,3],[0,3,5],[2,0,7]]`

`=[[3*5 + 2*0 + 1*2,3*2 + 2*3 + 1*0,3*3 + 2*5 + 1*7],[1*5 + 2*0 + 1*2,1*2 + 2*3 + 1*0,1*3 + 2*5 + 1*7],[2*5 + 3*0 + 0*2,2*2 + 3*3 + 0*0,2*3 + 3*5 + 0*7]]`

`=[[17,12,26],[7,8,20],[10,13,21]]`

`A×(B × C)=[[1,2,3],[0,1,0],[2,3,1]]×[[17,12,26],[7,8,20],[10,13,21]]`

`=[[1*17 + 2*7 + 3*10,1*12 + 2*8 + 3*13,1*26 + 2*20 + 3*21],[0*17 + 1*7 + 0*10,0*12 + 1*8 + 0*13,0*26 + 1*20 + 0*21],[2*17 + 3*7 + 1*10,2*12 + 3*8 + 1*13,2*26 + 3*20 + 1*21]]`

`=[[61,67,129],[7,8,20],[65,61,133]]`

`A×B=[[1,2,3],[0,1,0],[2,3,1]]×[[3,2,1],[1,2,1],[2,3,0]]`

`=[[1*3 + 2*1 + 3*2,1*2 + 2*2 + 3*3,1*1 + 2*1 + 3*0],[0*3 + 1*1 + 0*2,0*2 + 1*2 + 0*3,0*1 + 1*1 + 0*0],[2*3 + 3*1 + 1*2,2*2 + 3*2 + 1*3,2*1 + 3*1 + 1*0]]`

`=[[11,15,3],[1,2,1],[11,13,5]]`

`(A × B)×C=[[11,15,3],[1,2,1],[11,13,5]]×[[5,2,3],[0,3,5],[2,0,7]]`

`=[[11*5 + 15*0 + 3*2,11*2 + 15*3 + 3*0,11*3 + 15*5 + 3*7],[1*5 + 2*0 + 1*2,1*2 + 2*3 + 1*0,1*3 + 2*5 + 1*7],[11*5 + 13*0 + 5*2,11*2 + 13*3 + 5*0,11*3 + 13*5 + 5*7]]`

`=[[61,67,129],[7,8,20],[65,61,133]]`
 
9. Minor of a matrix
1. Minor(A)

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `"Minor"(A)` ...


`MINOR(A)=MINOR[[1,2,3],[0,1,0],[2,3,1]]`

`=[[|[1,0],[3,1]|,|[0,0],[2,1]|,|[0,1],[2,3]|],[|[2,3],[3,1]|,|[1,3],[2,1]|,|[1,2],[2,3]|],[|[2,3],[1,0]|,|[1,3],[0,0]|,|[1,2],[0,1]|]]`

`=[[1 × 1 - 0 × 3,0 × 1 - 0 × 2,0 × 3 - 1 × 2],[2 × 1 - 3 × 3,1 × 1 - 3 × 2,1 × 3 - 2 × 2],[2 × 0 - 3 × 1,1 × 0 - 3 × 0,1 × 1 - 2 × 0]]`

`=[[1 - 0,0 - 0,0 - 2],[2 - 9,1 - 6,3 - 4],[0 - 3,0 - 0,1 - 0]]`

`=[[1,0,-2],[-7,-5,-1],[-3,0,1]]`
10. Cofactor of a matrix
1. Cofactor(A)

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `"Cofactor"(A)` ...


`COFACTOR(A)=COFACTOR[[1,2,3],[0,1,0],[2,3,1]]`

`=[[+|[1,0],[3,1]|,-|[0,0],[2,1]|,+|[0,1],[2,3]|],[-|[2,3],[3,1]|,+|[1,3],[2,1]|,-|[1,2],[2,3]|],[+|[2,3],[1,0]|,-|[1,3],[0,0]|,+|[1,2],[0,1]|]]`

`=[[+(1 × 1 - 0 × 3),-(0 × 1 - 0 × 2),+(0 × 3 - 1 × 2)],[-(2 × 1 - 3 × 3),+(1 × 1 - 3 × 2),-(1 × 3 - 2 × 2)],[+(2 × 0 - 3 × 1),-(1 × 0 - 3 × 0),+(1 × 1 - 2 × 0)]]`

`=[[+(1 - 0),-(0 - 0),+(0 - 2)],[-(2 - 9),+(1 - 6),-(3 - 4)],[+(0 - 3),-(0 - 0),+(1 - 0)]]`

`=[[1,0,-2],[7,-5,1],[-3,0,1]]`
 
11. Trace of a matrix
1. Trace(A)

A = 
é 1  2  3 ù
ê 0  1  0 ú
ë 2  3  1 û

Find Matrix `Trace(A)` ...


`trace(A)=trace[[1,2,3],[0,1,0],[2,3,1]]`

`=1+1+1`

`=3`
 

 
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