https://file.aiquickdraw.com/mathbot/file/17292086412429ibgsi1x.jpg

To determine when the matrix \( A = \begin{bmatrix} 5 & k \\ k & 5 \end{bmatrix} \) is invertible, we need to check when its determinant is non-zero.

The determinant of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

\[
\text{det}(A) = ad - bc
\]

For the matrix \( A = \begin{bmatrix} 5 & k \\ k & 5 \end{bmatrix} \), we can calculate the determinant as:

\[
\text{det}(A) = (5)(5) - (k)(k) = 25 - k^2
\]

To ensure that \( A \) is invertible, we need:

\[
\text{det}(A) \neq 0
\]

This gives the inequality:

\[
25 - k^2 \neq 0
\]

\[
k^2 \neq 25
\]

\[
k \neq \pm 5
\]

Therefore, the matrix \( A \) is invertible for all values of \( k \) except \( k = \pm 5 \).

The correct answer is \( a) \, k \neq \pm 5 \).

Would you like further clarification or details on this process? Here are some related questions for deeper understanding:

1. What happens to the matrix if \( k = \pm 5 \)?
2. Can you generalize the determinant formula for other symmetric matrices like \( A \)?
3. How would you interpret the determinant geometrically?
4. How would the matrix change if the diagonal elements were different?
5. How can we find the inverse of matrix \( A \) if \( k \neq \pm 5 \)?

**Tip:** To check if a matrix is invertible, always verify that its determinant is non-zero.

Let A = [ [5, k], [k, 5] ], find all the value(s) of k such that A is invertible?

Learn how to determine the values of k that make the matrix A = [ [5, k], [k, 5] ] invertible by calculating its determinant and solving for k. This problem involves linear algebra concepts like determinants and matrix invertibility, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation