X 8.9. Let A = 3 -2 x² value(s) of x for which A is singular. Compute det(A). Use det(A) to determine the
X 8.9. Let A = 3 -2 x² value(s) of x for which A is singular. Compute det(A). Use det(A) to determine the
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Matrix and Determinant**
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**Example:**
Let \( A = \begin{bmatrix} x - 3 & \frac{x}{x^2} \\ -2 & x^2 \end{bmatrix} \). Compute det(A). Use det(A) to determine the value(s) of \( x \) for which \( A \) is singular.
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In this example, the task is to find the determinant of matrix \( A \) and use it to determine for what values of \( x \), matrix \( A \) is singular. A matrix is singular if its determinant is zero.
To compute the determinant of a 2x2 matrix \( \begin{bmatrix}a & b \\ c & d \end{bmatrix} \), use the formula:
\[ \text{det}(A) = ad - bc \]
In this case, matrix \( A \) is given by:
\[ A = \begin{bmatrix} x - 3 & \frac{x}{x^2} \\ -2 & x^2 \end{bmatrix} \]
So, applying the determinant formula:
\[ \text{det}(A) = (x - 3) \cdot (x^2) - \left( \frac{x}{x^2} \cdot (-2) \right) \]
Simplify the determinant equation and solve for \( x \) to find the values that make \( A \) singular.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f4d9a86-377e-4713-8398-ed20a45cabed%2F6c747683-e477-4dc8-9bc6-f9801d8eb3e2%2Foe2rzf9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix and Determinant**
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**Example:**
Let \( A = \begin{bmatrix} x - 3 & \frac{x}{x^2} \\ -2 & x^2 \end{bmatrix} \). Compute det(A). Use det(A) to determine the value(s) of \( x \) for which \( A \) is singular.
---
In this example, the task is to find the determinant of matrix \( A \) and use it to determine for what values of \( x \), matrix \( A \) is singular. A matrix is singular if its determinant is zero.
To compute the determinant of a 2x2 matrix \( \begin{bmatrix}a & b \\ c & d \end{bmatrix} \), use the formula:
\[ \text{det}(A) = ad - bc \]
In this case, matrix \( A \) is given by:
\[ A = \begin{bmatrix} x - 3 & \frac{x}{x^2} \\ -2 & x^2 \end{bmatrix} \]
So, applying the determinant formula:
\[ \text{det}(A) = (x - 3) \cdot (x^2) - \left( \frac{x}{x^2} \cdot (-2) \right) \]
Simplify the determinant equation and solve for \( x \) to find the values that make \( A \) singular.
![#### Determinants of Matrices
In this section, we will explore how to determine if a matrix is singular based on its determinant.
Consider the determinant expressed as follows:
\[ \text{det}(A) = x^3 - 3x^2 + 2x \]
This determinant is singular if the determinant function equals zero. In this case, the function \( x^3 - 3x^2 + 2x \) equals zero when \( x \) takes specific values.
#### Finding Singular Values
To determine the values of \( x \) that will make the matrix singular, we set the determinant to zero and solve for \( x \):
\[ x^3 - 3x^2 + 2x = 0 \]
Factoring the polynomial, we get:
\[ x(x^2 - 3x + 2) = 0 \]
\[ x(x - 1)(x - 2) = 0 \]
This equation will be satisfied if \( x \) equals:
\[ x = 0 \]
\[ x = 1 \]
\[ x = 2 \]
Therefore, the given matrix is singular when \( x = 0 \), \( x = 1 \), or \( x = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f4d9a86-377e-4713-8398-ed20a45cabed%2F6c747683-e477-4dc8-9bc6-f9801d8eb3e2%2Feidzb8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:#### Determinants of Matrices
In this section, we will explore how to determine if a matrix is singular based on its determinant.
Consider the determinant expressed as follows:
\[ \text{det}(A) = x^3 - 3x^2 + 2x \]
This determinant is singular if the determinant function equals zero. In this case, the function \( x^3 - 3x^2 + 2x \) equals zero when \( x \) takes specific values.
#### Finding Singular Values
To determine the values of \( x \) that will make the matrix singular, we set the determinant to zero and solve for \( x \):
\[ x^3 - 3x^2 + 2x = 0 \]
Factoring the polynomial, we get:
\[ x(x^2 - 3x + 2) = 0 \]
\[ x(x - 1)(x - 2) = 0 \]
This equation will be satisfied if \( x \) equals:
\[ x = 0 \]
\[ x = 1 \]
\[ x = 2 \]
Therefore, the given matrix is singular when \( x = 0 \), \( x = 1 \), or \( x = 2 \).
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