Linear Algebra: Matrix Inverses and Determinants PDF

# Section 2.2 ## Definition: An *n x n* matrix *A* is **invertible** or **non-singular** if there is an *n x n* matrix *C* satisfying *CA* = *I_n* and *AC* = *I_n*. If such a *C* exists it is unique and called the **inverse of A**. The inverse of *A* is denoted *A^-1...

# Section 2.2 ## Definition: An *n x n* matrix *A* is **invertible** or **non-singular** if there is an *n x n* matrix *C* satisfying *CA* = *I_n* and *AC* = *I_n*. If such a *C* exists it is unique and called the **inverse of A**. The inverse of *A* is denoted *A^-1*. If *A* is not invertible it is called **Singular**. ## Theorem: If *A* is an invertible *n x n* matrix then for every *b* ∈ ℝⁿ, *Ax = b* has the unique solution *x = A^-1*b. In particular, *A* has *n* pivots (one in every row and one in every column) and its *rref* is *I_n*. If *A*^-1 = *B* then *Ax = A(A^-1b) = (AA^-1)b = I_nb = b* If *Ax = b* then *A^-1(Ax) = A^-1b *(AA^-1)x = A^-1b* *I_nx = A^-1b* *x = A^-1b* ## Definition: The **determinant** of a 2 x 2 matrix $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$ is *det* $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$ = *ad - bc*. We will discuss determinants in general in Chapter 3. ## Theorem: If *A = $\begin{bmatrix}a & b \\ c & d \end{bmatrix}$* then *A* is invertible iff *det* *A* ≠ 0. If A is invertible *A^-1* = $\frac{1}{det A}\begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$ ## Example: Let *A* = **$\begin{bmatrix}3 & 4 \\ 2 & 1 \end{bmatrix}$**. Find *A^-1*, if it exists. *det A* = 3·1 - 2·4 = -2 (≠0 so *A* is invertible) *A^-1* = $\frac{1}{-2}\begin{bmatrix}1 & -4 \\ -2 & 3 \end{bmatrix}$ = **$\begin{bmatrix}-1/2 & 2 \\ 1 & -3/2\end{bmatrix}$** ## Example: Solve *x₁* + 2*x₂* = 5 3*x₁* + 4*x₂* = 6 This is *Ax = b* where *A* is as above. So Solution: *x* = *A^-1b* = **$\begin{bmatrix}-1/2 & 2 \\ 1 & -3/2\end{bmatrix}$** $\begin{bmatrix}5 \\ 6 \end{bmatrix}$ = **$\begin{bmatrix}-13/2 \\ 9/2 \end{bmatrix}$** ## Theorem: If *A, B* are invertible *n x n* matrices then - *A^-1* is invertible and *(A^-1)^-1* = *A* - *A^T* is invertible and *(A^T)^-1* = *(A^-1)^T* - *AB* is invertible and *(AB)^-1* = *B^-1*A^-1 * (reverse order!)* ## Definition: An *n x n* **elementary matrix** is a matrix obtained by performing a single elementary row operation on *I_n*. ## Example: Find the 2 x 2 elementary matrices corresponding to: Start with *I₂* = **$\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix}$** and apply each row operation - -4*R₁* + *R₂* -> *R₂* **$\begin{bmatrix}1 & 0 \\ -4 & 1 \end{bmatrix}$** - *R₁* ↔ *R₂* **$\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}$** - 3*R₂* -> *R₂* **$\begin{bmatrix}1 & 0 \\ 0 & 3 \end{bmatrix}$** Next, multiply **$\begin{bmatrix}4 & 5 & 6 \\ 1 & 2 & 3\end{bmatrix}$** on the left by each of these matrices. - **$\begin{bmatrix}1 & 0 \\ -4 & 1 \end{bmatrix}$** $\begin{bmatrix}4 & 5 & 6 \\ 1 & 2 & 3\end{bmatrix}$ = **$\begin{bmatrix}4 & 5 & 6 \\ 0 & -18 & -21\end{bmatrix}$** - **$\begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}$** $\begin{bmatrix}4 & 5 & 6 \\ 1 & 2 & 3\end{bmatrix}$ = **$\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix}$** - **$\begin{bmatrix}1 & 0 \\ 0 & 3 \end{bmatrix}$** $\begin{bmatrix}4 & 5 & 6 \\ 1 & 2 & 3\end{bmatrix}$ = **$\begin{bmatrix}4 & 5 & 6 \\ 3 & 6 & 9\end{bmatrix}$** ## Observation: Each elementary row operation has the same effect as multiplying on the left by the corresponding elementary matrix. Also, elementary matrices are invertible (since elementary row operations can be undone). ## Theorem: An *n x n* matrix *A* is invertible iff the *rref* of *A* is *I_n*. Additionally, any sequence of row operations that turns *A* into *I_n* will turn *I_n* into *A^-1*. Say *E₁*, ..., *E_k* are elementary matrices corresponding to row operations turning *A* into *I_n*. Then *E_k...E₁E_k...E₁A = I_n*. Set *B* = *E_k...E₁*. *B* is invertible, *BA* = *I_n*. *B*^-1*BA* = *B^-1I_n*. *I_nA* = *B^-1I_n*. *A* = *B^-1*, *A* is invertible. ## Algorithm for finding *A^-1*: If *A* invertible, the *rref* of [A *I_n*] is [I_n A^-1] ## Example: Find the inverse of *A* = **$\begin{bmatrix}1 & 0 & -2 \\ 0 & 1 & 1 \\ 2 & 3 & -1\end{bmatrix}$** if it exists. **$\begin{bmatrix}1 & 0 & -2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 2 & 3 & -1 & 0 & 0 & 1\end{bmatrix}$** *R3 - 2R1* -> *R3* **$\begin{bmatrix}1 & 0 & -2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 3 & 3 & -2 & 0 & 1\end{bmatrix}$** *R3 - 3R2* -> *R3* **$\begin{bmatrix}1 & 0 & -2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2 & -3 & 1\end{bmatrix}$** *R1 + 2R3* -> *R1* **$\begin{bmatrix}1 & 0 & -2 & -3 & -6 & 2 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2 & -3 & 1\end{bmatrix}$** *R2 - R3* -> *R2* **$\begin{bmatrix}1 & 0 & -2 & -3 & -6 & 2 \\ 0 & 1 & 1 & 2 & 4 & -1 \\ 0 & 0 & 0 & -2 & -3 & 1\end{bmatrix}$** *-1/2R3* -> *R3* **$\begin{bmatrix}1 & 0 & -2 & -3 & -6 & 2 \\ 0 & 1 & 1 & 2 & 4 & -1 \\ 0 & 0 & 0 & 1 & 3/2 & -1/2\end{bmatrix}$** *R1 + 3R3* -> *R1* **$\begin{bmatrix}1 & 0 & -2 & 0 & -3/2 & 1/2 \\ 0 & 1 & 1 & 2 & 4 & -1 \\ 0 & 0 & 0 & 1 & 3/2 & -1/2\end{bmatrix}$** *R2 - 2R3* -> *R2* **$\begin{bmatrix}1 & 0 & -2 & 0 & -3/2 & 1/2 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 3/2 & -1/2\end{bmatrix}$** *A^-1* = **$\begin{bmatrix}0 & -3/2 & 1/2 \\ 0 & 1 & 0 \\ 1 & 3/2 & -1/2 \end{bmatrix}$**