I need help with this both of these problems but this problem has to be done in the matrix form, can you do step by step with both problem I have already given the answer and question ### Example Data Points for Analysis**8.40 0**The first data point indicates a value of 8.40 associated with a secondary value of 0. This could represent a measurement, a time stamp, or a coded variable specific to the case study or dataset being analyzed.**8.41 5, 0**The second data point indicates a value of 8.41 with two associated secondary values: 5 and 0. These secondary values might represent different variables or conditions pertaining to the primary value. The comma may separate different qualifiers or codes related to the same primary measurement or observation.### Context for Educational PurposesIn an educational context, such data points can be used to explain the importance of precision in measurements and the relevance of secondary attributes or conditions in scientific data. The slight difference between the primary values (8.40 and 8.41) highlights the need for accurate data collection. Meanwhile, the secondary values (0 and 5, 0) prompt discussions about what these qualifiers might represent and their significance in the analysis.#### Graphical RepresentationWhile no graphs or diagrams are shown here, we can discuss potential ways to visualize this data:- **Line Graphs**: Represent the primary values along the y-axis and a sequence (or time) along the x-axis. Secondary values could be shown using different lines or markers.- **Scatter Plots**: Each data point can be plotted to show the relationship between the primary value and secondary values.- **Bar Charts**: Use grouped bar charts to represent primary values and the corresponding secondary values.These visualizations aid in better understanding the data distribution and potential correlations between the variables. ### Linear Algebra Problems and Solutions**Problem 8.40**The columns of a matrix \( \mathbf{A} \) are linearly dependent. Find \( \text{det}(\mathbf{A}) \).**Solution:**When the columns of a matrix \( \mathbf{A} \) are linearly dependent, it means that there exists some non-trivial linear combination of the columns that equals the zero vector. This implies that the determinant of \( \mathbf{A} \) is zero. Therefore,\[\text{det}(\mathbf{A}) = 0\]**Problem 8.41**Let \( \mathbf{A} \) and \( \mathbf{B} \) be \( 5 \times 5 \) matrices such that \( \mathbf{A}\mathbf{B} \) is invertible. Find the Rank(\(\mathbf{A}\)) and Nullity(\(\mathbf{B}\)).**Solution:**Given that \( \mathbf{A}\mathbf{B} \) is invertible implies that \( \mathbf{A}\mathbf{B} \) is of full rank. For \( 5 \times 5 \) matrices, this means that the rank of \( \mathbf{A}\mathbf{B} \) is 5.Since \( \mathbf{A}\mathbf{B} \) is invertible, both \( \mathbf{A} \) and \( \mathbf{B} \) must be of full rank because the product of the matrices is invertible only if both matrices are invertible. Hence, the ranks of both \( \mathbf{A} \) and \( \mathbf{B} \) must be 5.For a \( 5 \times 5 \) matrix:\[\text{Rank}(\mathbf{A}) = 5 \quad \text{and} \quad \text{Rank}(\mathbf{B}) = 5\]By the rank-nullity theorem:\[\text{Rank}(\mathbf{B}) + \text{Nullity}(\mathbf{B}) = 5\]Given that the rank of \( \mathbf{B} \) is 5:\[5 + \text{Nullity}(\mathbf{B}) = 5 \implies \text{Nullity}(\mathbf{B}) =

If the columns of matrix A are linearly dependent, it means that at least one column can be…

Answered: 8.40. The columns of a matrix A are…

8.40. The columns of a matrix A are linearly dependent. Find det(A). 8.41. Let A and B be 5 x 5 matrices such that AB is invertible. Find Rank(A) and Nullity(B).

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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