Find the multiplicative inverse of the matrix, if it exists.

Question

Find the multiplicative inverse of the matrix, if it exists. 

   1  0  6
−2  1  8
  5  0  2

Answer 1

I will show you the general method of calculating inverse of a matrix. It works with any square matrix. If the matrix does not have an inverse, you will get row with all zeros in the process.

 

Notation each row will be denoted with the index of the number of the row. For example, R₂ means row 2. We will replace a row in next matrix with a new row, that will be a linear combination of 2 rows. The notation R₂-3R₁ means that in the next matrix I will replace the row against you see that notation with the difference row 2 subtract (three times row 1).

 

In order to find the matrix inverse, we need to transform the current matrix into identity matrix. We will make the same transformations starting with an identity matrix. The resulting matrix will be the inverse of the starting matrix.

 

Starting matrix          | Transformation     | Identity matrix

transformations         |                            |transformations

 

 1    0    6                 |                            |  1     0     0

-2    1    20               |   2R₁ +R₂             |  0     1     0

5     0     2                |   5R₁ - R_{3               |   0      0      1}

 

(1)Next matrix :

 1    0    6                 |                            |   1     0    0

 0    1    20               |                            |    2    1    0

 0    0    28               |    1/28•R₃            |    5    0    -1

 

(2)Next matrix

 1     0    6                |    R₁-6R₃              |    1     0    0

 0     1   20               |    R₂-20R₃            |     2     1    0

 0     0     1               |                            |   5/28  0   -1/28

 

(3) Next matrix

 1     0    0                |                            | -1/14   0  3/14

 0     1    0                |                            |-11/7    1  5/7

 0     0    1                |                            |  5/28    0 -1/28

 

Calculations are show below for New matrix 1 and the transformations on the identity matrix.

2R₁+R₂                                          

a₂₂=2•0+1 =1                                

a₂₃=2•6+8=20                              

For identity matrix:

a₁₂=2•1+0=2

a₂₂=2•0+1=1

a₂₃=2•0+0=0

 

5R₁-R₃

a₃₂=5•0-0=0

a₃₃=5•6-2=28

For the inverse matrix:

a₃₁=5•1-0=5

a₃₂=5•0-0=0

a₃₃=5•0-1=-1

 

Note, a₂₁=a₃₁=0 (this is how we decided to make the transformations.

 

The rest of calculations are done the same way.

 

Suggestion: Use separate sheet of paper to write explicitly the calculations, including the members signs and work slowly to be able to calculate carefully. Check your work twice.

This is an algorithm. You can research and can find matrix calculator that calculates the inverse of a matrix for you, that will confirm your answer or show you made a mistake.