Without actually finding the inverse, determine if each matrix below has an inverse or not. Please explain each answer. ### Matrices in Linear Algebra#### Example MatricesThe following are examples of matrices typically used in linear algebra.**a)** \[\begin{bmatrix}5 & -2 & 10 & 3 \\3 & 4 & 8 & -6 \\2 & -4 & 3 & 6 \\0 & -6 & 0 & 9 \\\end{bmatrix}\]- This is a 4x4 matrix with varied positive and negative integers.**b)** \[\begin{bmatrix}-6 & 7 & 1 & 4 \\0 & 5 & 0 & 2 \\0 & 0 & -1 & 1 \\0 & 0 & 7 & 8 \\\end{bmatrix}\]- This is another 4x4 matrix. The structure shows that the lower left part of the matrix is mostly filled with zeros, which suggests a possible upper triangular form.**c)** Matrix $ A $\[A = \begin{bmatrix}1 & -5 & -4 \\0 & 3 & 4 \\-3 & 6 & 0 \\\end{bmatrix}\]- This is a 3x3 matrix. Each row contains both positive and negative integers. These matrices can be used to illustrate various operations and properties in linear algebra, such as addition, multiplication, determinants, and solving systems of linear equations.

Answered: Without actually finding the inverse,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Without actually finding the inverse, determine if each matrix below has an inverse or not. Please explain each answer.

### Matrices in Linear Algebra

#### Example Matrices

The following are examples of matrices typically used in linear algebra.

**a)**
\[
\begin{bmatrix}
5 & -2 & 10 & 3 \\
3 & 4 & 8 & -6 \\
2 & -4 & 3 & 6 \\
0 & -6 & 0 & 9 \\
\end{bmatrix}
\]

- This is a 4x4 matrix with varied positive and negative integers.

**b)**
\[
\begin{bmatrix}
-6 & 7 & 1 & 4 \\
0 & 5 & 0 & 2 \\
0 & 0 & -1 & 1 \\
0 & 0 & 7 & 8 \\
\end{bmatrix}
\]

- This is another 4x4 matrix. The structure shows that the lower left part of the matrix is mostly filled with zeros, which suggests a possible upper triangular form.

**c)** Matrix $ A $

\[
A =
\begin{bmatrix}
1 & -5 & -4 \\
0 & 3 & 4 \\
-3 & 6 & 0 \\
\end{bmatrix}
\]

- This is a 3x3 matrix. Each row contains both positive and negative integers.

These matrices can be used to illustrate various operations and properties in linear algebra, such as addition, multiplication, determinants, and solving systems of linear equations.

Expert Solution

Step 1: Problem(a)

Given matrix is $open square brackets table row 5 cell negative 2 end cell 10 3 row 3 4 8 cell negative 6 end cell row 2 cell negative 4 end cell 3 6 row 0 cell negative 6 end cell 0 9 end table close square brackets$

convert the above matrix into echelon form by using row operators

$R subscript 1 space rightwards arrow space 2 R subscript 2 minus R subscript 1$ $open square brackets table row 1 10 6 cell negative 15 end cell row 3 4 8 cell negative 6 end cell row 2 cell negative 4 end cell 3 6 row 0 cell negative 6 end cell 0 9 end table close square brackets$

$R subscript 2 space rightwards arrow space R subscript 2 minus 3 R subscript 1 space comma space space R subscript 3 space rightwards arrow space R subscript 3 minus 2 R subscript 1$ $open square brackets table row 1 10 6 cell negative 15 end cell row 0 cell negative 26 end cell cell negative 10 end cell 39 row 0 cell negative 24 end cell cell negative 9 end cell 36 row 0 cell negative 6 end cell 0 9 end table close square brackets$

$R subscript 3 space rightwards arrow space 26 R subscript 3 minus 24 R subscript 2 space end subscript space comma space space space space R subscript 4 space rightwards arrow space R subscript 3 minus 4 R subscript 4$ $open square brackets table row 1 10 6 cell negative 15 end cell row 0 cell negative 26 end cell cell negative 10 end cell 39 row 0 0 6 0 row 0 0 cell negative 9 end cell 0 end table close square brackets$

$R subscript 4 rightwards arrow 6 R subscript 4 plus 9 R subscript 3$ $open square brackets table row 1 10 6 cell negative 15 end cell row 0 cell negative 26 end cell cell negative 10 end cell 39 row 0 0 6 0 row 0 0 0 0 end table close square brackets$

The rank of the above matrix = Number of non zero rows in echelon form = 3

Rank < order or the matrix (4 x 4 )

hence the given matrix is singular ( det = 0 ), hence inverse matrix is not exist.