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Chapter 2: Q3Q (page 93)

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.

3.\(\left( {\begin{aligned}{*{20}{c}}5&0&0\\{ - 3}&{ - 7}&0\\8&5&{ - 1}\end{aligned}} \right)\)

Short Answer

Expert verified

The matrix \(\left( {\begin{aligned}{*{20}{c}}5&0&0\\{ - 3}&{ - 7}&0\\8&5&{ - 1}\end{aligned}} \right)\) is 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 identity matrix of the \(n \times n\) 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

At row one, multiply row one by \(\frac{1}{5}\).

\(\left( {\begin{aligned}{*{20}{c}}1&0&0\\{ - 3}&{ - 7}&0\\8&5&{ - 1}\end{aligned}} \right)\)

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

\(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 7}&0\\8&5&{ - 1}\end{aligned}} \right)\)

At row three, multiply row one by 8 and subtract it from row three.

\(\left( {\begin{aligned}{*{20}{c}}1&0&0\\0&{ - 7}&0\\0&5&{ - 1}\end{aligned}} \right)\)

03

Determine whether the matrix is invertible

A \(3 \times 3\) matrix has three pivot positions. It is invertible according to part (c) of the invertible matrix theorem.

Thus, the matrix \(\left( {\begin{aligned}{*{20}{c}}5&0&0\\{ - 3}&{ - 7}&0\\8&5&{ - 1}\end{aligned}} \right)\) is invertible.

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Most popular questions from this chapter

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{2}}\\{\bf{5}}&{{\bf{12}}}\end{aligned}} \right),{b_{\bf{1}}} = \left( {\begin{aligned}{*{20}{c}}{ - {\bf{1}}}\\{\bf{3}}\end{aligned}} \right),{b_{\bf{2}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}\\{ - {\bf{5}}}\end{aligned}} \right),{b_{\bf{3}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{2}}\\{\bf{6}}\end{aligned}} \right),\) and \({b_{\bf{4}}} = \left( {\begin{aligned}{*{20}{c}}{\bf{3}}\\{\bf{5}}\end{aligned}} \right)\).

  1. Find \({A^{ - {\bf{1}}}}\), and use it to solve the four equations \(Ax = {b_{\bf{1}}},\)\(Ax = {b_2},\)\(Ax = {b_{\bf{3}}},\)\(Ax = {b_{\bf{4}}}\)\(\)
  2. The four equations in part (a) can be solved by the same set of row operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix \(\left( {\begin{aligned}{*{20}{c}}A&{{b_{\bf{1}}}}&{{b_{\bf{2}}}}&{{b_{\bf{3}}}}&{{b_{\bf{4}}}}\end{aligned}} \right)\).

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)

Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{ - {\bf{4}}}\\{\bf{7}}&{ - {\bf{8}}}\end{aligned}} \right)\).

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

Suppose Tand U are linear transformations from \({\mathbb{R}^n}\) to \({\mathbb{R}^n}\) such that \(T\left( {U{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\) . Is it true that \(U\left( {T{\mathop{\rm x}\nolimits} } \right) = {\mathop{\rm x}\nolimits} \) for all x in \({\mathbb{R}^n}\)? Why or why not?
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