**Determinants of Matrices - Practice Problems**Consider the following determinants of matrices:Suppose $ \text{Det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = -10 $, $ \text{Det} \begin{bmatrix} a & -2 \\ c & 1 \end{bmatrix} = 10 $, and $ \text{Det} \begin{bmatrix} a & -2 \\ c & 0 \end{bmatrix} = -2 $.Calculate the following determinants:1. $ \text{Det} \begin{bmatrix} a & -2 + b \\ c & 1 + d \end{bmatrix} = \boxed{\phantom{}} $2. $ \text{Det} \begin{bmatrix} a & 0 \\ c & 1 \end{bmatrix} = \boxed{\phantom{}} $ ### Matrix Transformations and Their Effects on Geometric FiguresConsider the matrix $ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} $. This matrix represents a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$.#### Transformation of a Unit SquareWe will observe the effect of this transformation on a unit square whose vertices are at the points $ (0, 0) $, $ (1, 0) $, $ (1, 1) $, and $ (0, 1) $.**Vertices Before Transformation:**- $ (0, 0) $- $ (1, 0) $- $ (1, 1) $- $ (0, 1) $**Applying the Transformation**:Let’s apply the transformation matrix to each vertex:1. To $ (0, 0) $: \[ \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]2. To $ (1, 0) $: \[ \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \]3. To $ (1, 1) $: \[ \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]4. To $ (0, 1) $: \[ \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix}

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Answered: а — 2 a Suppose Det - 10, Det 10, and…

Math

Advanced Math

а — 2 a Suppose Det - 10, Det 10, and Det а —2 – 2. .cd] 1 -2 + b a Det = 1+d a Det

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

**Determinants of Matrices - Practice Problems**

Consider the following determinants of matrices:

Suppose $ \text{Det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = -10 $, $ \text{Det} \begin{bmatrix} a & -2 \\ c & 1 \end{bmatrix} = 10 $, and $ \text{Det} \begin{bmatrix} a & -2 \\ c & 0 \end{bmatrix} = -2 $.

Calculate the following determinants:

1. $ \text{Det} \begin{bmatrix} a & -2 + b \\ c & 1 + d \end{bmatrix} = \boxed{\phantom{}} $

2. $ \text{Det} \begin{bmatrix} a & 0 \\ c & 1 \end{bmatrix} = \boxed{\phantom{}} $

### Matrix Transformations and Their Effects on Geometric Figures

Consider the matrix $ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} $. This matrix represents a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$.

#### Transformation of a Unit Square

We will observe the effect of this transformation on a unit square whose vertices are at the points $ (0, 0) $, $ (1, 0) $, $ (1, 1) $, and $ (0, 1) $.

**Vertices Before Transformation:**
- $ (0, 0) $
- $ (1, 0) $
- $ (1, 1) $
- $ (0, 1) $

**Applying the Transformation**:
Let’s apply the transformation matrix to each vertex:

1. To $ (0, 0) $:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}
\begin{bmatrix} 0 \\ 0 \end{bmatrix} =
\begin{bmatrix} 0 \\ 0 \end{bmatrix}
\]

2. To $ (1, 0) $:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}
\begin{bmatrix} 1 \\ 0 \end{bmatrix} =
\begin{bmatrix} 1 \\ 0 \end{bmatrix}
\]

3. To $ (1, 1) $:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}
\begin{bmatrix} 1 \\ 1 \end{bmatrix} =
\begin{bmatrix} 1 \\ 3 \end{bmatrix}
\]

4. To $ (0, 1) $:
\[
\begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix}
\begin{bmatrix} 0 \\ 1 \end{bmatrix} =
\begin{bmatrix}

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