# Properties of Inverse Matrices

Properties of Inverse Matrices:

If *A* is nonsingular, then so is *A*^{ -1} and (*A*^{ -1}) ^{-1} = *A*

If *A* and B are nonsingular matrices, then *AB* is nonsingular and (*AB*)^{ -1} = *B*^{-1}* A*^{ -1}

If *A *is nonsingular then (*A*^{T})^{-1} = (*A*^{-1})^{T}

If *A* and *B* are matrices with *AB=I*_{n} then *A* and *B* are inverses of each other.

Notice that the fourth property implies that if *AB *= *I* then *BA *= *I*

Let *A*, *A*_{1} and *A*_{2} be *n×n* matrices, the following statements are true:

1. If *A*^{-1} = *B*, then *A* (col k of *B*) = e_{k}

2. If *A* has an inverse matrix, then there is only one inverse matrix.

3. If *A*_{1} and *A*_{2} have inverses, then *A*_{1} A_{2} has an inverse and (*A*_{1} A_{2})^{-1} = *A*_{1}^{-1 }A_{2}^{-1}

4. If *A* has an inverse, then x = *A*^{-1}d is the solution of *Ax = d* and this is the only solution.

5. The following are equivalent:

(i) *A* has an inverse.

(ii) det (*A*) is not zero.

(iii) *Ax = 0* implies x = 0.

If *c* is any non-zero scalar then *cA* is invertible and *(cA)*^{-1} = A^{-1}/c.

For n = 0, 1, 2…, *A*^{n} is invertible and *(A*^{n})^{-1} = A^{-n} = (A^{-1})^{n}.

If *A* is a square matrix and n > 0 then:

*A*^{-n} = (A^{-1})^{n
}

**Example 1**: Compute *A*^{-3} for the matrix:

**Solution:** First of all, we need to find the inverse of the given matrix. The method to find the inverse is only applicable for 2 × 2 matrices.

Steps are as follows:

*[1] Interchange leading diagonal elements:*

-7 → 2

2 → -7

*[2] Change signs of the other 2 elements:*

-3 → 3

4 → -4

*[3] Find the determinant |A|*

*[4] Multiply result of [2] by 1/|A|*

Now:

**Example 2**: Given the matrix *A* verify that the indicated matrix is in fact the inverse.

Solution:

To verify that we do in fact have the inverse we’ll need to check the condition

*AA*^{-1} = A^{-1}A = I

Now we check whether *AA*^{-1} = A^{-1}A = I:

Hence the required is verified.

**Example 3**: Let *A* be the 2 × 2 matrix,

Show that *A* has no inverse.

Solution: An inverse for *A* must be a 2 × 2 matrix.

Let such that *AB = BA = I*. If such a matrix *B* exists, it must satisfy the following equation:

The preceding equation requires that:

*a + 2c = 1* and *3a + 6c = 0*

which is clearly impossible, so we can conclude that A has no inverse.