7.6.1: Inverse Matrices (Exercises)

SECTION 2.4 PROBLEM SET: INVERSE MATRICES

In problems 1- 2, verify that the given matrices are inverses of each other.

$\left[\begin{array}{ll} 7 & 3 \\ 2 & 1 \end{array}\right]\left[\begin{array}{rr} 1 & -3 \\ -2 & 7 \end{array}\right]$

$\left[\begin{array}{ccc} 1 & -1 & 0 \\ 1 & 0 & -1 \\ 2 & 3 & -4 \end{array}\right]\left[\begin{array}{ccc} 3 & -4 & 1 \\ 2 & -4 & 1 \\ 3 & -5 & 1 \end{array}\right]$

In problems 3- 6, find the inverse of each matrix by the row-reduction method.

$\left[\begin{array}{rr} 3 & -5 \\ -1 & 2 \end{array}\right]$

$\left[\begin{array}{lll} 1 & 0 & 2 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{array}\right]$

SECTION 2.4 PROBLEM SET: INVERSE MATRICES

In problems 5 - 6, find the inverse of each matrix by the row-reduction method.

$\left[\begin{array}{ccc} 1 & 1 & -1 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right]$

$\left[\begin{array}{lll} 1 & 1 & 1 \\ 3 & 1 & 0 \\ 1 & 1 & 2 \end{array}\right]$

Problems 7 -10: Express the system as $AX = B$; then solve using matrix inverses found in problems 3 - 6.

$\begin{array}{l} 3 x-5 y=2 \\ -x+2 y=0 \end{array}$

$\begin{aligned} x+\quad 2 z &=8 \\ y+4 z &=8 \\ z &=3 \end{aligned}$

SECTION 2.4 PROBLEM SET: INVERSE MATRICES

Problems 9 -10: Express the system as $AX = B$; then solve using matrix inverses found in problems 3 - 6.

$\begin{aligned} x+y-z &=2 \\ x+ z&=7 \\ 2 x+y+z &=13 \end{aligned}$	$\begin{array}{l} x+y+z=2 \\ 3 x+y=7 \\ x+y+2 z=3 \end{array}$
Why is it necessary that a matrix be a square matrix for its inverse to exist? Explain by relating the matrix to a system of equations.	Suppose we are solving a system $AX = B$ by the matrix inverse method, but discover $A$ has no inverse. How else can we solve this system? What can be said about the solutions of this system?