Inverses of Matrices with Formula
Inverses of Matrices with Formula - How it Works - Video
Formula
Inverse Identities:
For matrix to have an identity, the matrix must be a square. For example, it must be 2 x 2, 3 x 3, 4 x 4, 5 x 5, and so on.
Each identity is the diagonal from top left to bottom right contains ones and the rest are zeros.
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.
Formula - Different Way
Inverse Identities:
For the 2 x 2 matrix instead of having det(A), we have written what det(A) equals, and that is a * d - b * c.
Now for the 3 x 3 matrix, R = row and C = column. For instance, if we draw a line through row 1 and column 1, then we use the remaining elements to find the determinant for C1,1. Next, if we draw a line through row 2 and column 1, then we use the remaining elements to find the determinant for C1,2. Next, if we draw a line through row 3 and column 1, then we use the remaining elements to find the determinant for C1,3. Then, we continue down the line to find more determinants.
This formula is little bit much in my opinion. Gauss-Jordan elimination is probably your best bet to find the inverse of a 3 x 3 matrix other than a calculator.
Example 1
Example 1:
We have labeled a = 4, b = -3, c = 5, and d = 2. Now we can substitute those numbers into the formula to find our inverse.
The det(A) is 23 because 4 * 2 - (-3) * 5 ==> 8 +15 ==> 23. So our we are going multiply the numbers after rearranging them according to the formula. So our final answer is for the first row is 2/23 and 3/23, and our second row is -5/23 and 4/23.
Example 2 - Part 1
Example 2 - Part 1:
Our first step is find the determinant of the matrix since we have a 3 x 3. It is a bit more complicated. Now remember we can break down the matrix by using any row. Look for the ones that have the most 0s. After breaking it down using the first row, our determinant of matrix is 100.
Example 2 - Part 2
Example 2 - Part 2:
Now our next is label each number with the corresponding letter so that we don't get confused. Once we find the determinants of each location, and multiply by 1/100. Then our first row is -13/50 and 21/50 and 3/50. Our second row is 3/50 and -1/50 and -7/50. Our third row is 21/100 and -7/100 and -1/100.
This is a lot of remembering and arithmetic. I think that the best method would be using the Gauss-Jordan Elimination method to find the inverse. But you get to choose, so give it go if you feel like.