Inverse of a Matrix

An n x n matrix A is said to have an inverse provided there exists an n x n matrix B such that AB = BA = In. We call B the inverse of A and denote it as A-1. Thus, AA-1 = A-1A = In. In this case, A is also called nonsingular.

Example.

$A = \left(\begin{array}{cc}4&3\\3&2\end{array}\right)$
$A^{-1} = \left(\begin{array}{cc}-2&3\\3&-4\end{array}\right)$

$AA^{-1} = \left(\begin{array}{cc}4&3\\3&2\end{array}\right) $$\left(\begin{array}{cc}-2&3\\3&-4\end{array}\right) = \left(\begin{array}{cc}1&0\\0&1\end{array}\right) and A^{-1} = \left(\begin{array}{cc}-2&3\\3&-4\end{array}\right)$$ \left(\begin{array}{cc}4&3\\3&2\end{array}\right) =$$\left(\begin{array}{cc}1&0\\0&1\end{array}\right)$

Theorem 1

The inverse of a matrix, if it exists, is unique

Theorem 2

If A and B are both nonsingular n x n matrices (i.e. invertible), then AB is nonsingular and (AB)-1 = B-1A-1.

Corollary 1

If A1, A2, ..., Ar are n x n nonsingular matrices, then A1A2...Ar is nonsingular an (A1A2...Ar)-1 = Ar-1Ar-1-1...A1-1.

Theorem 3

If A is a nonsingular matrix, then A-1 is nonsingular and (A-1)-1 = A.

Theorem 4

If A is a nonsingular matrix, then AT is nonsingular and (A-1)T = (AT)-1.

List of Equivalences

The following are equivalent for an n x n matrix A:
1. A is nonsingular.
2. Ax = 0 has only the trivial solution.
3. A is row(column equivalent to I. (The reduced row echelon form of A is I).
4. The linear system Ax = b has a unique solution for every n x 1 matrix b.
5. A is a product of elementary matrices.

Methods for determining the inverse of a matrix

1. Shortcut for determining the inverse of a 2 x 2 matrix

If $A = \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ then the inverse of matrix A can be found using:
$A^{-1} = \frac{1}{detA}\left(\begin{array}{cc}d&-b\\-c&a\end{array}\right) = \frac{1}{ad - bc}\left(\begin{array}{cc}d&-b\\-c&a\end{array}\right)$

Example

$A = \left(\begin{array}{cc}1&2\\3&4\end{array}\right)$
$detA = ad - bc = 1 \times 4 - 2 \times 3 = -2$
$A^{-1} = \frac{1}{-2}\left(\begin{array}{cc}4&-2\\-3&1\end{array}\right)$
$A^{-1} = \left(\begin{array}{cc}-2&1\\\frac{3}{2}&\frac{-1}{2}\end{array}\right)$

2. Augmented Matrix Method (Can be used for any n x n matrix)

Use Gauss-Jordan Elimination to transform [ A | I ] into [ I | A-1 ].

$\left(\begin{array}{ccc|ccc}a&b&c&1&0&0\\d&e&f&0&1&0\\g&h&i&0&0&1\end{array}\right)$

Perform elementary row operations to transform the A matrix into the identity matrix. Whatever row operations are carried out on the A matrix, should also be carried out on the augmented Identity matrix. The final state of the matrix on the left is the inverse of the A matrix.

Theorem 1

An n x n matrix A is singular if and only if A is row equivalent to a matrix B that has a row of zeros. (That is, the reduced row echelon form of A has a row of zeros.)

Theorem 2

If A and B are n x n matrices such that AB = I, then BA = I. Thus B = A-1.

Example. Find A-1 of
$A = \left(\begin{array}{cc}1&2\\3&4\end{array}\right)$

$\left(\begin{array}{cc|cc}1&2&1&0\\3&4&0&1\end{array}\right)\longrightarrow\left(\begin{array}{cc|cc}1&2&1&0\\0&-2&-3&1\end{array}\right)\longrightarrow\left(\begin{array}{cc|cc}1&2&1&0\\0&1&\frac{3}{2}&\frac{-1}{2}\end{array}\right)\longrightarrow \left(\begin{array}{cc|cc}1&0&-2&1\\0&1&\frac{3}{2}&\frac{-1}{2}\end{array}\right)$

so
$A^{-1} = \left(\begin{array}{cc}-2&1\\\frac{3}{2}&\frac{-1}{2}\end{array}\right)$

3. Adjoint Matrix Method (Can be used for any n x n matrix)

$A^{-1} = \frac{1}{detA}(adjoint(A)) = \frac{1}{detA}(cofactor(A^{T}))$

Example. Find A-1 of
$A = \left(\begin{array}{ccc}1&2&3\\0&4&5\\1&0&6\end{array}\right)$
$detA = 4 \times (6 - 3) - 5 \times (0 - 2) = 12 + 10 = 22$
$adjoint(A) = \left(\begin{array}{ccc}24&-12&-2\\5&3&-5\\-4&2&4\end{array}\right)$
$A^{-1} = \frac{1}{22}\left(\begin{array}{ccc}24&-12&-2\\5&3&-5\\-4&2&4\end{array}\right) = \left(\begin{array}{ccc}\frac{12}{11}&\frac{-6}{11}&\frac{-1}{11}\\\frac{5}{22}&\frac{3}{22}&\frac{-5}{11}\\\frac{-2}{11}&\frac{1}{11}&\frac{2}{11}\end{array}\right)$ 