What is the inverse of this matrix?

Question

What is the inverse of this matrix?

_ _
|1 5 2 |
|1 1 7 |
|0 -3 4 |
_ _
Find the inverse. If u can, plz tell me how u did it. I already have an answer , i want to check if i am doing it right. I dont know if the procedure is correct.

Answer 1

One way to solve this is by using the augmented matrix with an identity matrix as a part of it (on the right side).
| 1 5 2 : 1 0 0 |
| 1 1 7 : 0 1 0 |
| 0 -3 4 : 0 0 1 |
You can use elimination to make an identity matrix on the left side, and when you do, whatever's on the right is the inverse of the original matrix.

Work in columns, left to right. In solving, your original matrix already has a "1" element in the first column, first row. In column 1, row 2, you need to get rid of the 1.
| 1 5 2 : 1 0 0 |
| 0 -4 5 : -1 1 0 | (Row 2 - Row 1)
| 0 -3 4 : 0 0 1 |
In column 1, row 3, you already have a zero, so this column is done. Move on to column 2.

To get a "1" in column 2, row 2 (always start with your ones in the main diagonal, then work on the zeroes in the rest of the column), multiply -1/4 by the second row.
| 1 5 2 : 1 0 0 |
| 0 1 -5/4 : 1/4 -1/4 0 | ((-1/4)*Row 2)
| 0 -3 4 : 0 0 1 |
Now get zeroes in the other elements of that column.
| 1 0 33/4 : -1/4 5/4 0 | (Row 1 - 5*Row 2)
| 0 1 -5/4 : 1/4 -1/4 0 |
| 0 0 1/4 : 3/4 -3/4 1 | (Row 3 + 3*Row 2)

On to column 3. Multiply Row 3 by 4 to make your main diagonal correct.
| 1 0 33/4 : -1/4 5/4 0 |
| 0 1 -5/4 : 1/4 -1/4 0 |
| 0 0 1 : 3 -3 4 | (4*Row 3)
Now finish column 3 and you're done.
| 1 0 0 : -25 26 -33 | (Row 1 - (33/4)*Row 3)
| 0 1 0 : 4 -4 5 | (Row 2 + (5/4)*Row 3)
| 0 0 1 : 3 -3 4 |

The inverse matrix is
| -25 26 -33 |
| 4 -4 5 |
| 3 -3 4 |

Answer 2

we detect the inverse of the matrix by technique of the co-ingredient approach and For the first row: the cofactor of two is a million x 0 - 4 x (-a million) = 0 + 4 = 4 the cofactor of four is -a million x 0 - a million x (-a million) = 0 + a million = a million the cofactor of one million is -a million x 4 - a million x a million = -4 -a million = -5 For the 2d row: the cofactor of -a million is 4 x 0 - 4 x a million = 0 -4 = -4 the cofactor of one million is two x 0 - a million x (a million) = 0 - a million = -a million the cofactor of -a million is two x 4 - a million x a million = 8 -4 = 4 For the 0.33 row: the cofactor of one million is 4 x-a million - a million x (a million) = -4 -a million = -5 the cofactor of four is two x -a million - a million x (-a million) =-2+ a million = -a million the cofactor of 0 is two x a million - -a million x 4 = 2 +4 = 6 So the cofactor matrix obtained is 4 a million -5 -4 -a million 4 -5 -a million 6 replacing the signs and indicators + - + - + - + - + we get 4 -a million -5 4 -a million -4 -5 a million 6 Transposing this matrix and dividing by technique of determinant of A we get ( the price of determinant A = 2(1x0 - 4 x -a million) - 4(-1x0 - 1x -a million) + a million(-a million x 4- 1x a million) = 2 (4) - 4(a million) + a million(-5) = 8 -4 -5 = -a million -4 -4 5 a million a million -a million 5 4 -6 subsequently our answer

Answer 3

Write down the matrix and beside it the 'I' matrix

1 5 2 | 1 0 0
1 1 7 | 0 1 0
0 -3 4 | 0 0 1

now make the left matrix the 'I' matrix by adding / subracting rows, you are also allowed to multiply one row with some number.

Every operation you do on the left matrix, you will also do with the 'I' matrix.

In ther end when left is the 'I', the right matrix will be the inverse of your matrix.

Answer 4

A good check is to multiply the original matrix by its inverse. If you did it correctly, you'll get the identity matrix.

Answer 5

|-25 26 -33|
|4 -4 -5|
|3 -3 4|