In A & B, evaluate the determinant of the given matrix by inspection.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
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In A & B, evaluate the determinant of the given matrix by inspection.
![This image depicts a 4x4 matrix, which is a key concept in linear algebra. The matrix is presented in a standard rectangular array of numbers, arranged in rows and columns. Here is the matrix:
\[
\begin{pmatrix}
0 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 4 & 3 & 0 \\
1 & 2 & 3 & 8
\end{pmatrix}
\]
Each element of the matrix is a number arranged in a specific row and column. This particular matrix consists of the following elements:
- Element in the first row and first column is 0.
- Element in the first row and second column is 0.
- Element in the first row and third column is 0.
- Element in the first row and fourth column is 0.
- Element in the second row and first column is 1.
- Element in the second row and second column is 2.
- Element in the second row and third column is 0.
- Element in the second row and fourth column is 0.
- Element in the third row and first column is 0.
- Element in the third row and second column is 4.
- Element in the third row and third column is 3.
- Element in the third row and fourth column is 0.
- Element in the fourth row and first column is 1.
- Element in the fourth row and second column is 2.
- Element in the fourth row and third column is 3.
- Element in the fourth row and fourth column is 8.
Matrices are used in various mathematical and engineering disciplines to represent linear transformations, solve systems of linear equations, and many other applications. Understanding the structure and properties of matrices is fundamental in these areas.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87643561-f7f8-47c3-82b0-7735ae0706bf%2F43cb570d-2046-440b-b54a-50bad1aa7485%2F6k4n3co_processed.png&w=3840&q=75)
Transcribed Image Text:This image depicts a 4x4 matrix, which is a key concept in linear algebra. The matrix is presented in a standard rectangular array of numbers, arranged in rows and columns. Here is the matrix:
\[
\begin{pmatrix}
0 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 4 & 3 & 0 \\
1 & 2 & 3 & 8
\end{pmatrix}
\]
Each element of the matrix is a number arranged in a specific row and column. This particular matrix consists of the following elements:
- Element in the first row and first column is 0.
- Element in the first row and second column is 0.
- Element in the first row and third column is 0.
- Element in the first row and fourth column is 0.
- Element in the second row and first column is 1.
- Element in the second row and second column is 2.
- Element in the second row and third column is 0.
- Element in the second row and fourth column is 0.
- Element in the third row and first column is 0.
- Element in the third row and second column is 4.
- Element in the third row and third column is 3.
- Element in the third row and fourth column is 0.
- Element in the fourth row and first column is 1.
- Element in the fourth row and second column is 2.
- Element in the fourth row and third column is 3.
- Element in the fourth row and fourth column is 8.
Matrices are used in various mathematical and engineering disciplines to represent linear transformations, solve systems of linear equations, and many other applications. Understanding the structure and properties of matrices is fundamental in these areas.
![Here is our given matrix, represented in a standard 3x3 format.
\[
\begin{bmatrix}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]
This is a diagonal matrix where all the diagonal elements are equal to 2 and all non-diagonal elements are zero. Diagonal matrices are a special type of square matrix with the property that all off-diagonal elements are zero.
In this specific matrix:
- The element in the first row and first column (position [1,1]) is 2.
- The element in the second row and second column (position [2,2]) is 2.
- The element in the third row and third column (position [3,3]) is 2.
All other elements are 0.
This matrix is often used in linear transformations and can represent scaling in 3D space where each axis is scaled by the factor of 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87643561-f7f8-47c3-82b0-7735ae0706bf%2F43cb570d-2046-440b-b54a-50bad1aa7485%2Ffy1xyd_processed.png&w=3840&q=75)
Transcribed Image Text:Here is our given matrix, represented in a standard 3x3 format.
\[
\begin{bmatrix}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]
This is a diagonal matrix where all the diagonal elements are equal to 2 and all non-diagonal elements are zero. Diagonal matrices are a special type of square matrix with the property that all off-diagonal elements are zero.
In this specific matrix:
- The element in the first row and first column (position [1,1]) is 2.
- The element in the second row and second column (position [2,2]) is 2.
- The element in the third row and third column (position [3,3]) is 2.
All other elements are 0.
This matrix is often used in linear transformations and can represent scaling in 3D space where each axis is scaled by the factor of 2.
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