Linear Algebra **Problem 8: Solve the following.**(a) Find a 4x4 matrix whose determinant is 7.(b) Use row reduction to find the value of $ k $ that makes $\text{det}(A) = 0$.Matrix $ A $ is given as:\[ A = \begin{bmatrix} 1 & -2 & 4 \\ 0 & 1 & 5 \\ -3 & 6 & -3k \end{bmatrix} \]**Explanation:**- Part (a): Construct a 4x4 matrix and ensure calculations verify that the determinant equals 7. - Part (b): Apply row reduction methods on the given 3x3 matrix to determine the value of $ k $ which results in a determinant of zero.**Note:** Row reduction involves using elementary row operations to transform the matrix into an upper triangular form, which simplifies the process of calculating the determinant.

First we have to find a 4×4 matrix whose determinant is 7. Concept: since, we know that Determinant…

Answered: 8. Solve the following. (a) Find a 4 x…

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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Linear Algebra

**Problem 8: Solve the following.**

(a) Find a 4x4 matrix whose determinant is 7.

(b) Use row reduction to find the value of $ k $ that makes $\text{det}(A) = 0$.

Matrix $ A $ is given as:

\[
A = \begin{bmatrix}
1 & -2 & 4 \\
0 & 1 & 5 \\
-3 & 6 & -3k
\end{bmatrix}
\]

**Explanation:**

- Part (a): Construct a 4x4 matrix and ensure calculations verify that the determinant equals 7.

- Part (b): Apply row reduction methods on the given 3x3 matrix to determine the value of $ k $ which results in a determinant of zero.

**Note:** Row reduction involves using elementary row operations to transform the matrix into an upper triangular form, which simplifies the process of calculating the determinant.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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