2.4.1: Inverse Matrices (Exercises)

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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?