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Unless otherwise specified, assume that all matrices in these exercises are \(n \times n\). Determine which of the matrices in Exercises 1-10 are invertible. Use a few calculations as possible. Justify your answer.

5. \(\left[ {\begin{aligned}{*{20}{c}}0&3&{ - 5}\\1&0&2\\{ - 4}&{ - 9}&7\end{aligned}} \right]\)

Short Answer

Expert verified

The matrix \(\left[ {\begin{aligned}{*{20}{c}}0&3&{ - 5}\\1&0&2\\{ - 4}&{ - 9}&7\end{aligned}} \right]\) is not invertible.

Step by step solution

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01

State the invertible matrix theorem

Let Abe a square \(n \times n\) matrix. Then the following statements are equivalent.

For a given A, all these statements are either true or false.

  1. Ais an invertible matrix.
  2. Ais row equivalent to the \(n \times n\) matrix identity matrix.
  3. Ahas n pivot positions.
  4. The equation Ax = 0 has only a trivial solution.
  5. The columns of A form a linearly independent set.
  6. The linear transformation \(x \mapsto Ax\) is one-to-one.
  7. The equation \(Ax = b\) has at least one solution for each b in \({\mathbb{R}^n}\).
  8. The columns of Aspan \({\mathbb{R}^n}\).
  9. The linear transformation \(x \mapsto Ax\) maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\).
  10. There is an \(n \times n\) matrix Csuch that CA = I.
  11. There is an \(n \times n\) matrix Dsuch that DA = I.
  12. \({A^T}\) is an invertible matrix.
02

Apply the row operation

Interchange rows one and row two.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\{ - 4}&{ - 9}&7\end{aligned}} \right]\)

At row three, multiply row one by 4 and add it to row three.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\0&{ - 9}&{15}\end{aligned}} \right]\)

At row three, multiply row two by 3 and add it to row three.

\(\left[ {\begin{aligned}{*{20}{c}}1&0&2\\0&3&{ - 5}\\0&0&0\end{aligned}} \right]\)

03

Determine whether the matrix is invertible

According to part (b) of the invertible matrix theorem, the matrix is not row equivalent to the identity matrix.

Thus, the matrix \(\left[ {\begin{aligned}{*{20}{c}}0&3&{ - 5}\\1&0&2\\{ - 4}&{ - 9}&7\end{aligned}} \right]\) is not invertible.

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Most popular questions from this chapter

Use partitioned matrices to prove by induction that for \(n = 2,3,...\), the \(n \times n\) matrices \(A\) shown below is invertible and \(B\) is its inverse.

\[A = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\1&1&0&{}&0\\1&1&1&{}&0\\ \vdots &{}&{}& \ddots &{}\\1&1&1& \ldots &1\end{array}} \right]\]

\[B = \left[ {\begin{array}{*{20}{c}}1&0&0& \cdots &0\\{ - 1}&1&0&{}&0\\0&{ - 1}&1&{}&0\\ \vdots &{}& \ddots & \ddots &{}\\0&{}& \ldots &{ - 1}&1\end{array}} \right]\]

For the induction step, assume A and Bare \(\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrices, and partition Aand B in a form similar to that displayed in Exercises 23.

Show that if the columns of Bare linearly dependent, then so are the columns of AB.

Explain why the columns of an \(n \times n\) matrix Aare linearly independent when Ais invertible.

Let Abe an invertible \(n \times n\) matrix, and let \(B\) be an \(n \times p\) matrix. Explain why \({A^{ - 1}}B\) can be computed by row reduction: If\(\left( {\begin{aligned}{*{20}{c}}A&B\end{aligned}} \right) \sim ... \sim \left( {\begin{aligned}{*{20}{c}}I&X\end{aligned}} \right)\), then \(X = {A^{ - 1}}B\).

If Ais larger than \(2 \times 2\), then row reduction of \(\left( {\begin{aligned}{*{20}{c}}A&B\end{aligned}} \right)\) is much faster than computing both \({A^{ - 1}}\) and \({A^{ - 1}}B\).

Assume \(A - s{I_n}\) is invertible and view (8) as a system of two matrix equations. Solve the top equation for \({\bf{x}}\) and substitute into the bottom equation. The result is an equation of the form \(W\left( s \right){\bf{u}} = {\bf{y}}\), where \(W\left( s \right)\) is a matrix that depends upon \(s\). \(W\left( s \right)\) is called the transfer function of the system because it transforms the input \({\bf{u}}\) into the output \({\bf{y}}\). Find \(W\left( s \right)\) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 15.

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