- We’ll be looking at square matrices only in these cases
- Reminder that for matrices A, B, AB does not necessarily equal BA
For a matrix A, the its inverse A−1 should have the following property: A−1A=AA−1=In.
Ex. Determine if
AB=(2−35−7),=(−7352) are inverses of each other.
AB=(2−35−7)(−7352)=(1001)=I2.
Therefore, B=A−1 and A=B−1.
If we had the equation 5x=1, we would solve it by multiplying both sides by 51, the multiplicative inverse of 5. We can use an analogous operation to solve systems of linear equations.
Suppose that A is an invertible matrix. Then the equation Ax=b has a unique solution given by x=A−1b. We can prove this using properties of matrix multiplication:
AxA−1(Ax)(A−1A)xInxx=b=A−1b=A−1b=A−1b=A−1b.
In the case of a 2x2 matrix A=(acbd), if ad−bc=0, then A has an inverse, which is given by
A−1=ad−bc1(d−c−ba).
Ex. Is (5438) invertible? (5)(8)−(3)(4)=28=0, so it is invertible.
You can also find the inverse of a matrix by putting an identity matrix to the right of it and row-reducing the left matrix into the identity matrix. Afterwards, the right matrix will be the inverse of the original left matrix.
(Any invertible matrix will be row-equivalent to the identity matrix. If you can’t row-reduce a matrix into the identity matrix, then it’s non-invertible.)
Ex. Find the inverse of A.
A=→→→→→→→⎝⎛01410−3238⎠⎞⎝⎛01410−3238100010001⎠⎞⎝⎛10401−3328010100001⎠⎞⎝⎛10001−332−401010−4001⎠⎞⎝⎛10001032201310−4001⎠⎞⎝⎛100010321012310−20021⎠⎞⎝⎛1000103010−22314−20−121⎠⎞⎝⎛100010001−29−22374−2−23−121⎠⎞
Now that the left side is I3, the right side must be A−1, so
A−1=⎝⎛−29−22374−2−23−121⎠⎞.