Inverse Matrices

Definition. A square matrix $A$ is called nonsingular if $det A \neq = 0$ , and singular (sometimes degenerate) if $det A = 0$ .

The Inverse Matrix

Definition. A matrix $A^{- 1}$ is called the inverse of a square matrix $A$ if

A \cdot A^{- 1} = A^{- 1} \cdot A = E .

A matrix $A$ for which $A^{- 1}$ exists is called invertible. The inverse matrix has the same order as $A$ .

Theorem (uniqueness). If an inverse matrix exists, it is unique.

Proof. Suppose $A$ has two inverses $A_{1}^{- 1}$ and $A_{2}^{- 1}$ . Then

A_{1}^{- 1} = A_{1}^{- 1} E = A_{1}^{- 1} (A A_{2}^{- 1}) = (A_{1}^{- 1} A) A_{2}^{- 1} = E A_{2}^{- 1} = A_{2}^{- 1} .

The Invertibility Criterion

Theorem (Invertibility criterion). A matrix $A$ is invertible if and only if it is nonsingular:

\exists A^{- 1} ⟺ det A \neq = 0.

Proof. Necessity. Suppose $A$ is invertible. From $A A^{- 1} = E$ we get $det A \cdot det A^{- 1} = det E = 1$ , hence $det A \neq = 0$ (and along the way, $det A^{- 1} = \frac{1}{d e t A}$ ).

Sufficiency. If $det A \neq = 0$ , the inverse matrix is given by an explicit formula via the adjugate matrix (below).

The Adjugate Matrix and the Formula for the Inverse

Definition. The adjugate matrix $A$ is the transpose of the matrix of cofactors of the entries of $A$ :

A = (A_{ij})^{T} = A_{11} A_{12} ⋮ A_{1 n} A_{21} A_{22} ⋮ A_{2 n} \dots \dots ⋱ \dots A_{n 1} A_{n 2} ⋮ A_{nn}

(note the transposition: position $(i, j)$ holds $A_{j i}$ ).

Multiplying $A$ by $A$ and using theorems 11 and 12 (expansion of the determinant and the vanishing of "foreign" cofactor sums), we get

A \cdot A = A \cdot A = det A \cdot E .

Hence, when $det A \neq = 0$ :

A^{- 1} = \frac{1}{det A} A .

Algorithm for Finding the Inverse Matrix

Compute $det A$ . If $det A = 0$ , there is no inverse; if $det A \neq = 0$ , it exists.
Find all the cofactors $A_{ij}$ .
Assemble the matrix of cofactors $B = (A_{ij})$ .
Transpose: the adjugate matrix is $A = B^{T}$ .
Compute $A^{- 1} = \frac{1}{det A} A$ .

Cost estimate. For an $n \times n$ matrix, one must compute $n^{2}$ cofactors — minors of order $(n - 1)$ . So for large $n$ this method is expensive, and in practice the inverse is more often found by the method of elementary transformations (below).

Formula for a $2 \times 2$ Matrix

For $A = (a c b d)$ with $det A = a d - b c \neq = 0$ :

A^{- 1} = \frac{1}{a d - b c} (d - c - b a) .

Example: $A = (2111)$ , $det A = 1$ , so $A^{- 1} = (1 - 1 - 1 2)$ (check: $A A^{- 1} = E$ ).

Properties of the Inverse Matrix

$E^{- 1} = E$ (since $E \cdot E = E$ ).
$det (A^{- 1}) = \frac{1}{det A}$ .
$(A^{- 1})^{- 1} = A$ (from $A A^{- 1} = A^{- 1} A = E$ ).
$(A^{T})^{- 1} = (A^{- 1})^{T}$ . Proof. $(A^{- 1})^{T} A^{T} = (A A^{- 1})^{T} = E^{T} = E$ and $A^{T} (A^{- 1})^{T} = (A^{- 1} A)^{T} = E$ .
$(A \cdot B)^{- 1} = B^{- 1} \cdot A^{- 1}$ (the order reverses!). Proof. $(B^{- 1} A^{- 1}) (A B) = B^{- 1} (A^{- 1} A) B = B^{- 1} E B = E$ , and similarly $(A B) (B^{- 1} A^{- 1}) = E$ .
$(λ A)^{- 1} = \frac{1}{λ} A^{- 1}$ for $λ \neq = 0$ .

Application: Matrix Equations

The inverse matrix makes it possible to solve matrix equations. Let $det A \neq = 0$ .

Type I: $A X = B$ — multiply on the left by $A^{- 1}$ :

A^{- 1} (A X) = A^{- 1} B \Rightarrow E X = A^{- 1} B \Rightarrow X = A^{- 1} B .

Type II: $X A = B$ — multiply on the right by $A^{- 1}$ :

(X A) A^{- 1} = B A^{- 1} \Rightarrow X E = B A^{- 1} \Rightarrow X = B A^{- 1} .

The order of multiplication matters because of non-commutativity: in Type I the factor is on the left, in Type II on the right.

The Method of Elementary Transformations (Gauss–Jordan)

Elementary row transformations are: (I) multiplying a row by a number $λ \neq = 0$ ; (II) swapping two rows; (III) adding to a row another row multiplied by a number. Each such transformation is equivalent to multiplying on the left by the corresponding elementary matrix.

Theorem. Any nonsingular matrix can be reduced to the identity by elementary row transformations: $T_{k} \dots T_{2} T_{1} \cdot A = E$ . Denoting $T = T_{k} \dots T_{1}$ , we have $T A = E$ , that is, $T = A^{- 1}$ . So the same transformations, applied to $E$ , give $A^{- 1}$ .

In practice, one writes the augmented matrix $(A ∣ E)$ and uses row transformations to bring the left half to $E$ :

(A ∣ E) elementary row transformations (E ∣ A^{- 1}) .

Example

A = 210 0 - 1 0 101, det A = 1 \cdot (- 1)^{3 + 3} 21 0 - 1 = - 2 \neq = 0 \Rightarrow \exists A^{- 1} .

210 0 - 1 0 101100010001 R_{1} \leftrightarrow R_{2} 120 - 1 00 011010100001 R_{2} - 2 R_{1} 100 - 1 20 011010 1 - 2 0 001

R_{2} \cdot \frac{1}{2} 100 - 1 10 0 \frac{1}{2} 1 0 \frac{1}{2} 0 1 - 1 0 001 R_{2} - \frac{1}{2} R_{3} R_{1} + R_{2} 100010001 \frac{1}{2} \frac{1}{2} 0 0 - 1 0 - \frac{1}{2} - \frac{1}{2} 1

\Rightarrow A^{- 1} = \frac{1}{2} \frac{1}{2} 0 0 - 1 0 - \frac{1}{2} - \frac{1}{2} 1 .

(Check: $A \cdot A^{- 1} = E$ .)

Related works