How to Find the Inverse of a 3 x 3 Matrix

Like in multiplication where if we multiply by the multiplicative identity the answer will be 1. For example 5 * (1/5) is 1. When we multiply, a matrix and it's inverse matrix the resultant matrices will just have the number 1 inside.

Example 1 - Step 1

Example 1 - Step 1:

Here we had R₁ - 2 * R₂replace R₂ and the reason is that we want all the numbers beneath a_1,1or 2 in this case to be 0.

The main number that we are focusing on here is the 1 underneath the 2. We multiplied the second row by 2 so that we can get 2 - 2 = 0 for the element a_2,1. If we get any other 1s or 0s, then that is great. If not, more arithmetic.

Example 1 - Step 2

Example 1 - Step 2:

Here we had 2*R₁ - * R₃replace R₃ and the reason is that we want the number in a_3,1or 4 in this case to be 0.

The main number that we are focusing on here is the 4 underneath the 0. We multiplied the first row by 2 so that we can get 4 - 4 = 0 for the element a_3,1. If we get any other 1s or 0s, then that is great. If not, more arithmetic.

Example 1 - Step 3

Example 1 - Step 3:

Here we had 2*R₃replace R₃ and the reason is that we want the number in a₂_,3or -2 in this case to be 0.

The main number that we are focusing on here is the -2 underneath the 0. We multiplied the third row by 2 so that we can get 6. We want to add or subtract row 2 and row 3 to that a_2,3is 0. In order to do that we want to find the least common denominator of |-2| or a_2,3 and 3 or a₃_,3. The least common denominator of 2 and 3 is 6. Now we multiplied row 3 by 2 to 6 for a_3,3.

Example 1 - Step 4

Example 1 - Step 4:

Here we had 3*R₂+ R₃replace R₂ and the reason is that we want the number in a_2,3or -2 in this case to be 0.

The main number that we are focusing on here is the -2 underneath the 0. We multiplied the second row by 3 so that we can add the third row and we get - 6 + 6 = 0 for the element a_2,3.

Example 1 - Step 5

Example 1 - Step 5:

Here we had 3*R₂ - R₃replace R₃ and the reason is that we want the number in a_2,2or -12 in this case to be 0.

The main number that we are focusing on here is the -12 underneath the -6. We multiplied the second row by 3 so that we can subtract the third row and we get -12 - (-12) = 0 for the element a_3,2.

Example 1 - Step 6

Example 1 - Step 6:

Here we had 2*R₂replace R₂ and the reason is that we want the number in a_2,2or -6 in this case to be 1.

The main number that we are focusing on here is the -6 underneath the -4. We multiplied the second row by 2 so that we can subtract the first row in the next step and we get -6 * 2 = -12 for the element a₂_,2.

Example 1 - Step 7

Example 1 - Step 7:

Here we had 3*R₁ - R₂replace R₁ and the reason is that we want the number in a₁_,2or -4 in this case to be 0.

The main number that we are focusing on here is the -4 above the -12. We multiplied the first row by 3 so that we can subtract the second row and we get -12 - (-12) = 0 for the element a_2,2.

Example 1 - Step 8

Example 1 - Step 8:

Here we divide each row by the multiplicative inverse of number in a_1,₁, a₂_,2, and a_3,3.

So 1/6 times the first row because 1/6 * 6 = 1.

So -1/12 times the first row because -1/12 * (-12) = 1.

So -1/6 times the first row because -1/6 * (-6) = 1.

Inverse

Inverse:

After all of that, we have found the inverse.

I would recommend showing all of steps for this because if you make one mistake at the very beginning it will take forever to find the mistake without any work. So if you don't show it, it might be easier to start over.

Now this is just one way to get the inverse. I would recommend leaving the divide so get the identity as the last step so you don't have fractions, but that is up to you.

Multiply Inverse by the Constants - Step 9

Multiply Inverse by the Constants - Step 9:

Here we multiply the inverse by the constants to find the solution to our system of equations.