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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

2. Solve linear equation of any number of variables using
1. Inverse Matrix method
2. Cramer's Rule method
3. Gauss Elimination method
4. Gauss Sield 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]]`
 

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