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Section 4.4 Invertible Matrices (M4)

Activity 4.4.1.

Is the matrix \(\left[\begin{array}{ccc} 2 & 3 & 1 \\ -1 & -4 & 2 \\ 0 & -5 & 5 \end{array}\right]\) invertible? Give a reason for your answer.

Observation 4.4.1.

An \(n\times n\) matrix \(A\) is invertible if and only if \(\RREF(A) = I_n\text{.}\)

Activity 4.4.2.

Let \(T:\IR^2\to\IR^2\) be the bijective linear map defined by \(T\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 2x -3y \\ -3x + 5y\end{array}\right]\text{,}\) with the inverse map \(T^{-1}\left(\left[\begin{array}{c}x\\y\end{array}\right]\right)=\left[\begin{array}{c} 5x+ 3y \\ 3x + 2y\end{array}\right]\text{.}\)

(a)

Compute \((T^{-1}\circ T)\left(\left[\begin{array}{c}-2\\1\end{array}\right]\right)\text{.}\)

(b)

If \(A\) is the standard matrix for \(T\) and \(A^{-1}\) is the standard matrix for \(T^{-1}\text{,}\) find the \(2\times 2\) matrix

\begin{equation*} A^{-1}A=\left[\begin{array}{ccc}\unknown&\unknown\\\unknown&\unknown\end{array}\right]. \end{equation*}
Observation 4.4.2.

\(T^{-1}\circ T=T\circ T^{-1}\) is the identity map for any bijective linear transformation \(T\text{.}\) Therefore \(A^{-1}A=AA^{-1}=I\) is the identity matrix for any invertible matrix \(A\text{.}\)

Exercises 4.4.1 Exercises

1.

Explain why the matrix \(M= \left[\begin{array}{cccc} 1 & -1 & -2 & 2 \\ -2 & 3 & 3 & -5 \\ 2 & -5 & -1 & 7 \\ 2 & -5 & -1 & 7 \end{array}\right] \) is or is not invertible.

Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -2 & 2 \\ -2 & 3 & 3 & -5 \\ 2 & -5 & -1 & 7 \\ 2 & -5 & -1 & 7 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 1 \\ 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\(M\) is not invertible.
2.

Explain why the matrix \(N= \left[\begin{array}{cccc} 1 & -1 & -7 & -4 \\ 1 & 0 & -3 & -1 \\ -1 & -1 & 0 & -2 \\ 0 & 2 & 4 & 6 \end{array}\right] \) is or is not invertible.

Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -7 & -4 \\ 1 & 0 & -3 & -1 \\ -1 & -1 & 0 & -2 \\ 0 & 2 & 4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\(N\) is not invertible.
3.

Explain why the matrix \(N= \left[\begin{array}{cccc} 1 & -2 & 8 & 7 \\ 0 & 1 & -2 & -1 \\ 1 & -1 & 6 & 7 \\ 1 & 0 & 4 & 5 \end{array}\right] \) is or is not invertible.

Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & 8 & 7 \\ 0 & 1 & -2 & -1 \\ 1 & -1 & 6 & 7 \\ 1 & 0 & 4 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 4 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \end{equation*}
\(N\) is not invertible.
4.

Explain why the matrix \(P= \left[\begin{array}{cccc} 1 & 4 & -6 & -4 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -1 & -6 \\ 0 & 4 & -7 & -1 \end{array}\right] \) is or is not invertible.

Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -6 & -4 \\ 0 & 1 & -1 & -2 \\ 0 & 2 & -1 & -6 \\ 0 & 4 & -7 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \end{equation*}
\(P\) is invertible.
5.

Explain why the matrix \(C= \left[\begin{array}{cccc} 1 & 1 & 3 & -2 \\ 1 & 2 & 6 & -7 \\ -2 & -1 & -2 & -4 \\ -3 & -2 & -8 & 8 \end{array}\right] \) is or is not invertible.

Answer
\begin{equation*} \operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 3 & -2 \\ 1 & 2 & 6 & -7 \\ -2 & -1 & -2 & -4 \\ -3 & -2 & -8 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \end{equation*}
\(C\) is invertible.

Additional exercises available at checkit.clontz.org.