This document discusses inverses of matrices. It defines an invertible matrix as a square matrix A that has an inverse matrix B such that AB and BA are the identity matrix. It also defines singular and non-singular matrices. Theorems are provided to determine if a 2x2 matrix is invertible based on its determinant, and to solve systems of equations using the inverse matrix. Elementary matrices from row operations on the identity matrix are introduced. An algorithm for finding the inverse of an invertible matrix using row operations on the augmented matrix [A|I] is also given.

1. 2.2 Inverse of a Matrix

2. Definition [Invertible Matrix]
An nxn matrix A is said to be invertible if there is an nxn matrix B such
that
AB=BA=I
in which case B is an inverse of A and we write B=A-1

3. Definition [Singular and Non Singular
Matrices]
A matrix that is not invertible is called a singular matrix. A matrix that is
invertible is called non singular matrix.

4. Example: The matrix B=
−2 1
1.5 −0.5
is an inverse
of A =
1 2
3 4
.
Solution:

5. Remark:
Statement: If a square matrix has an inverse then it is unique.
proof: Let A be an invertible matrix. If B be the inverse of the matrix A then
by definition,
AB = BA = I …………(i)
Suppose if possible, let C be another inverse of A, then again by definition,
AC = CA = I………….(ii)
Now, C = CI = CAB from (i)
= IB from (ii)
= B, thus B is unique.

6. Theorem 4
Let A=
𝑎 𝑏
𝑐 𝑑
, ad − bc ≠ 0, then A is invertible and
𝐴−1 =
1
𝑎𝑑−𝑏𝑐
𝑑 −𝑏
−𝑐 𝑎
. If ad-bc=0, then A is invertible.
Proof:

8. Definition [Determinant]
Let A=
𝑎 𝑏
𝑐 𝑑
. The quantity ad-bc is called the determinant of A. It is
denoted by detA or 𝐴 or ∆𝐴.
Thus, 𝐴 =ad-bc

9. Note:
1. Theorem 4 guarantees that a 2x2 matrix A is invertible if and only if
detA≠ 0.
2. A matrix A is singular if detA=0 and it is non singular if detA ≠ 0.

10. Example: Find the inverse of A=
2 4
3 5
.
Solution:

11. Theorem 5
If A is an invertible nxn matrix, then for each b in Rn, the equation Ax=b
has the unique solution x=A-1b.
Proof:

12. Example: Use the inverse of the matrix to
solve the system, 2x + 4y = 6 , 3x + 5y = 8.
Solution:

13. Theorem 6
a. If A is an invertible matrix, then A-1 is invertible and (A-1)-1=A.
b. If A and B are nxn invertible matrices, then so is AB and the inverse
of AB is the product of the inverses of A and B the reverse order.
That is (AB)-1=B-1A-1.
c. If A is an invertible matrix, then so is AT, and the inverse of AT is the
transpose of A-1. That is (AT)-1=(A-1)T.
Proof:

15. Elementary Matrices
A matrix that is obtained by performing a single elementary row
operation in an identity matrix is called an elementary matrix.
e.g. 𝐸1 =
1 0 0
0 1 0
−4 0 1
, is an elementary matrix which is obtained by
performing the operation R3→R3 – 4R1

16. Algorithm for finding A-1
Let A be and invertible matrix.
1. Row reduce the augmented matrix [A | I].
2. If A is row equivalent to I, then [A | I] is row equivalent to [I | A-1]
i.e. [A | I] ~ … ~ [I | A-1].
3. Otherwise , A does not have an inverse.

17. Example