verify that det(A B)=det(B A) and determine whether the equality det (A+B)=det(A)+det(B) holds.

Name: Problem 6, Chapter 2: Elementary Linear Algebra: Applications Version
Uploaded: 2021-06-18T06:07:27-08:00
Duration: 3 min 11 s
Channel: Anthony Ramos
Description: verify that det(A B)=det(B A) and determine whether the equality det (A+B)=det(A)+det(B) holds. A=[ -1 8 2 1 0 -1 -2 2 2 ] and B=[ 2 -1 -4 1 1 3 0 3 -1 ]

step 1 Hello there. So for this exercise we have these two matrices A and V and we need to check the fo

Question

verify that $\operatorname{det}(A B)=\operatorname{det}(B A)$ and determine whether the equality det $(A+B)=\operatorname{det}(A)+\operatorname{det}(B)$ holds. $$A=\left[\begin{array}{rrr} -1 & 8 & 2 \\ 1 & 0 & -1 \\ -2 & 2 & 2 \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{rrr} 2 & -1 & -4 \\ 1 & 1 & 3 \\ 0 & 3 & -1 \end{array}\right]$$

   verify that $\operatorname{det}(A B)=\operatorname{det}(B A)$ and determine whether the equality det $(A+B)=\operatorname{det}(A)+\operatorname{det}(B)$ holds.
$$A=\left[\begin{array}{rrr}
-1 & 8 & 2 \\
1 & 0 & -1 \\
-2 & 2 & 2
\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{rrr}
2 & -1 & -4 \\
1 & 1 & 3 \\
0 & 3 & -1
\end{array}\right]$$

Elementary Linear Algebra: Applications Version

Howard Anton, Chris… 11th Edition

Chapter 2, Problem 6

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Solved by Expert Anthony Ramos

06/18/2021

Step 1

The product of A and B is given by: \[AB = \left[\begin{array}{ccc} 6 & 15 & 26 \\ 2 & -4 & -3 \\ -2 & 10 & 12 \end{array}\right]\] The determinant of this matrix is -66. Show more…

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verify that $\operatorname{det}(A B)=\operatorname{det}(B A)$ and determine whether the equality det $(A+B)=\operatorname{det}(A)+\operatorname{det}(B)$ holds. $$A=\left[\begin{array}{rrr} -1 & 8 & 2 \\ 1 & 0 & -1 \\ -2 & 2 & 2 \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{rrr} 2 & -1 & -4 \\ 1 & 1 & 3 \\ 0 & 3 & -1 \end{array}\right]$$

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Key Concepts

Determinant Multiplicative Property

This property states that the determinant of a product of two matrices is equal to the product of their determinants, that is, det(AB) = det(A)·det(B). As a consequence, even though matrix multiplication is generally not commutative, the equality det(AB) = det(BA) holds because both expressions equal det(A)·det(B). This concept is fundamental in linear algebra when relating products of transformations to the scaling of volume or areas in the respective spaces.

Non-Linearity of the Determinant Function

Unlike the multiplicative property, the determinant function does not distribute over matrix addition. This means that in general, the determinant of a sum of matrices, det(A+B), is not equal to the sum of their determinants, det(A)+det(B). This non-linearity highlights the intrinsic complexity of the determinant as a function and its sensitivity to the interactions between the entries of the matrices involved.

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