What is an identity matrix? An identity matrix is a matrix that has 1’s on the major diagonal (from top left to bottom right) and 0’s everywhere else. Here’s a couple:
[ 1 0]
[ 0 1]
[1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
As you can see, each matrix has 1’s going from the top left corner to the bottom right corner, with 0’s everywhere else. The great thing about identity matrices is that if you multiply square matrix by an identity matrix, the multiplication commutes:
[ 2 3 ] × [ 1 0 ] = [ 2 3 ]
[ 3 8 ] × [ 0 1 ] = [ 3 8 ]
[ 1 0 ] × [ 2 3 ] = [ 2 3 ]
[ 0 1 ] × [ 3 8 ] = [ 3 8 ]
Inverse matrices, though, are a little trickier. Although the inverse of number a is: 1/2, but we don’t just do that to every element in a matrix and say that it’s inverted.
Instead, what we do requires a little bit of algebra. One fact that helps us find the inverse matrix is that if square matrix A is multiplied by it’s inverse, square matrix A-1, then we get an identity matrix of the corresponding dimensions.
So let’s get started by trying to find the inverse of matrix: [ 7 4 ]
[ 2 3 ]
So we know that matrix: [ 7 4 ], if multiplied by some unknown matrix: [ w x ], is going
[ 2 3 ] [ y z ]
to equal the identity matrix: [ 1 0 ]. So we can write that as an equation:
[ 0 1 ]
[ 7 4 ] × [ w x ] = [ 1 0 ]
[2 3 ] × [ z y ] = [ 0 1 ]
Based on what we know about matrix multiplication, we can also write that [ 7 4 ]
multiplied by [ w x ] is equal to: [ 7w + 4y 7x + 4z ] [ 2 3 ]
[ z y ] [ 2w + 3y 2x + 3z ]
Since we also know that each one of those expressions is supposed to stand for a particular in the corresponding positions in the identity matrix, we can write them as equations and solve them:
7w + 4y = 1 7x + 4z = 0
2w + 3y = 0 2x + 3z = 1
Continuing on, we know that we can subtract or add the equations so that we can find out the value of a variable. Let’s do that with 7w + 4y = 1 and 2w + 3y = 0:
((7w + 4y = 1) ×3) 21w + 12y = 3
((2w + 3y = 0) ×-4)- 8w + 12y = 0
13w + 0y = 3
From here, it’s easy to do the rest. w = 3/13, and then we just substitute 3/13 into the equation, 2w + 3y to solve for y.
We get that 6/13 + 3y = 0. So from here, we can gather that 3y = –6/13 => y = –6/13 × 1/3 => –2/13.
So w = 6/13, and y = –2/13. We repeat the process with the other two equations, and the end result is that z = 7/13 and x = –4/13.
At last, we finally have the answer: the inverse of matrix [ 7 4 ] is [6/13 –4/13]
[ 2 3 ] [-2/13 7/13]
But there’s a much simpler formula for finding the inverse of the matrix. Here it is:
|a b| ) × [ d -b ] = [ a b]-1
(-1 ÷ |c d|) × [ -c a ] = [c d]
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