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Chapter 2: Q2Q (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.

2.\(\left( {\begin{aligned}{*{20}{c}}{ - 4}&6\\6&{ - 9}\end{aligned}} \right)\)

Short Answer

Expert verified

The matrix \(\left( {\begin{aligned}{*{20}{c}}{ - 4}&6\\6&{ - 9}\end{aligned}} \right)\) is not invertible.

Step by step solution

01

State the condition for an invertible matrix

Step 1: State the condition for an invertible matrix

02

Determine whether the matrix is invertible

It is not immediately apparent that the columns of the matrix \(\left( {\begin{aligned}{*{20}{c}}{ - 4}&6\\6&{ - 9}\end{aligned}} \right)\) are multiples. If the determinant is zero, the fastest check is likely to be determinant. The matrix is not invertible according to theorem 4 since its determinant is zero.

Thus, the matrix \(\left( {\begin{aligned}{*{20}{c}}{ - 4}&6\\6&{ - 9}\end{aligned}} \right)\) is not invertible.

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Most popular questions from this chapter

If A, B, and X are \(n \times n\) invertible matrices, does the equation \({C^{ - 1}}\left( {A + X} \right){B^{ - 1}} = {I_n}\) have a solution, X? If so, find it.

In Exercise 9 mark each statement True or False. Justify each answer.

9. a. In order for a matrix B to be the inverse of A, both equations \(AB = I\) and \(BA = I\) must be true.

b. If A and B are \(n \times n\) and invertible, then \({A^{ - {\bf{1}}}}{B^{ - {\bf{1}}}}\) is the inverse of \(AB\).

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ab - cd \ne {\bf{0}}\), then A is invertible.

d. If A is an invertible \(n \times n\) matrix, then the equation \(Ax = b\) is consistent for each b in \({\mathbb{R}^{\bf{n}}}\).

e. Each elementary matrix is invertible.

Use partitioned matrices to prove by induction that the product of two lower triangular matrices is also lower triangular. [Hint: \(A\left( {k + 1} \right) \times \left( {k + 1} \right)\) matrix \({A_1}\) can be written in the form below, where \[a\] is a scalar, v is in \({\mathbb{R}^k}\), and Ais a \(k \times k\) lower triangular matrix. See the study guide for help with induction.]

\({A_1} = \left[ {\begin{array}{*{20}{c}}a&{{0^T}}\\0&A\end{array}} \right]\).

Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).

Prove the Theorem 3(d) i.e., \({\left( {AB} \right)^T} = {B^T}{A^T}\).

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